Groups with symmetric cross-cap number 3 to 65 ............................................. The symmetric cross-cap number of a finite group G is the smallest genus of all the compact non-orientable surfaces on which G acts faithfully as a group of automorphisms. This parameter was considered by Tom Tucker in the 1980s as a variant of the "symmetric genus" (for actions on orientable surfaces) and defined by Coy May in the 1990s). The list below gives all the finite groups that have symmetric cross-cap number between 3 and 65 inclusive, together with the signature types for the action(s) of the group on (non-orientable) surfaces of the given smallest possible genus. The notation "Group of order n # k" stands for the kth group of order n in the "Small Groups Database" available in GAP and MAGMA. This list was created with the help of the MAGMA system, in March 2010. Marston Conder June 2013 ........................................................................... Groups with symmetric cross-cap number 3 Number of different groups with symmetric cross-cap number 3 = 0 ........................................................................... Groups with symmetric cross-cap number 4 Group of order 24 # 13 acting with signature (0; +; [6]; {(2)}) Group of order 48 # 48 acting with signature (0; +; [-]; {(2,4,6)}) Number of different groups with symmetric cross-cap number 4 = 2 ........................................................................... Groups with symmetric cross-cap number 5 Group of order 9 # 2 acting with signature (1; -; [3,3]; {-}) Group of order 18 # 3 acting with signature (0; +; [6]; {(3)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 18 # 4 acting with signature (0; +; [2]; {(3,3)}) Group of order 20 # 3 acting with signature (0; +; [4]; {(5)}) Group of order 36 # 9 acting with signature (0; +; [4]; {(3)}) Group of order 36 # 10 acting with signature (0; +; [-]; {(2,6,6)}) ... and with signature (0; +; [-]; {(2,2,2,3)}) Group of order 72 # 40 acting with signature (0; +; [-]; {(2,4,6)}) Group of order 120 # 34 acting with signature (0; +; [-]; {(2,4,5)}) Number of different groups with symmetric cross-cap number 5 = 8 ........................................................................... Groups with symmetric cross-cap number 6 Group of order 8 # 4 acting with signature (0; +; [4,4]; {(1)}) Group of order 16 # 3 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) ... and with signature (0; +; [-]; {(1),(2)}) Group of order 16 # 6 acting with signature (0; +; [-]; {(1),(2)}) Group of order 16 # 8 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 16 # 13 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 16 # 14 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 32 # 27 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 32 # 43 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 80 # 49 acting with signature (0; +; [5]; {(2)}) Group of order 120 # 35 acting with signature (0; +; [-]; {(2,3,10)}) Group of order 160 # 234 acting with signature (0; +; [-]; {(2,4,5)}) Number of different groups with symmetric cross-cap number 6 = 11 ........................................................................... Groups with symmetric cross-cap number 7 Group of order 12 # 1 acting with signature (0; +; [3,4]; {(1)}) Group of order 24 # 8 acting with signature (0; +; [-]; {(2,2,3,4)}) Group of order 36 # 3 acting with signature (0; +; [9]; {(2)}) Group of order 72 # 15 acting with signature (0; +; [-]; {(2,4,9)}) Number of different groups with symmetric cross-cap number 7 = 4 ........................................................................... Groups with symmetric cross-cap number 8 Group of order 24 # 5 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 24 # 10 acting with signature (0; +; [-]; {(1),(2)}) Group of order 48 # 38 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 56 # 11 acting with signature (0; +; [7]; {(2)}) Group of order 504 # 156 acting with signature (0; +; [-]; {(2,3,7)}) Number of different groups with symmetric cross-cap number 8 = 5 ........................................................................... Groups with symmetric cross-cap number 9 Group of order 21 # 1 acting with signature (1; -; [3,3]; {-}) Group of order 30 # 1 acting with signature (0; +; [10]; {(3)}) Group of order 30 # 2 acting with signature (0; +; [6]; {(5)}) Group of order 42 # 1 acting with signature (0; +; [2,3]; {(1)}) Group of order 60 # 8 acting with signature (0; +; [-]; {(2,6,10)}) Group of order 168 # 42 acting with signature (0; +; [3]; {(4)}) ... and with signature (0; +; [-]; {(3,3,4)}) Group of order 336 # 208 acting with signature (0; +; [-]; {(2,3,8)}) Number of different groups with symmetric cross-cap number 9 = 7 ........................................................................... Groups with symmetric cross-cap number 10 Group of order 16 # 2 acting with signature (0; +; [4,4]; {(1)}) Group of order 16 # 4 acting with signature (0; +; [4,4]; {(1)}) Group of order 16 # 9 acting with signature (0; +; [4,4]; {(1)}) Group of order 16 # 10 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 24 # 3 acting with signature (0; +; [3,3]; {(1)}) Group of order 32 # 5 acting with signature (0; +; [-]; {(1),(2)}) Group of order 32 # 6 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) ... and with signature (0; +; [-]; {(1),(2)}) Group of order 32 # 7 acting with signature (0; +; [-]; {(1),(2)}) Group of order 32 # 9 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 32 # 11 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 32 # 17 acting with signature (0; +; [-]; {(1),(2)}) Group of order 32 # 19 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 32 # 28 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 32 # 34 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 32 # 42 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 32 # 46 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 32 # 49 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 48 # 29 acting with signature (0; +; [-]; {(2,2,3,3)}) ... and with signature (0; +; [3]; {(2,2)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 48 # 31 acting with signature (0; +; [12]; {(2)}) Group of order 48 # 33 acting with signature (0; +; [3]; {(2,2)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 48 # 50 acting with signature (0; +; [3]; {(2,2)}) Group of order 64 # 128 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 64 # 134 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 64 # 138 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 64 # 190 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 96 # 70 acting with signature (0; +; [6]; {(2)}) Group of order 96 # 187 acting with signature (0; +; [-]; {(2,4,12)}) Group of order 96 # 193 acting with signature (0; +; [-]; {(2,2,2,3)}) Group of order 96 # 227 acting with signature (0; +; [-]; {(3,4,4)}) ... and with signature (0; +; [-]; {(2,2,2,3)}) Group of order 192 # 955 acting with signature (0; +; [-]; {(2,4,6)}) Number of different groups with symmetric cross-cap number 10 = 30 ........................................................................... Groups with symmetric cross-cap number 11 Group of order 18 # 5 acting with signature (0; +; [3,6]; {(1)}) Group of order 27 # 3 acting with signature (1; -; [3,3]; {-}) Group of order 36 # 13 acting with signature (0; +; [-]; {(2,2,3,6)}) Group of order 54 # 5 acting with signature (0; +; [6]; {(3)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 54 # 8 acting with signature (0; +; [2]; {(3,3)}) Group of order 108 # 15 acting with signature (0; +; [4]; {(3)}) Group of order 108 # 17 acting with signature (0; +; [-]; {(2,6,6)}) ... and with signature (0; +; [-]; {(2,2,2,3)}) Group of order 216 # 87 acting with signature (0; +; [-]; {(2,4,6)}) Number of different groups with symmetric cross-cap number 11 = 8 ........................................................................... Groups with symmetric cross-cap number 12 Group of order 20 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 40 # 5 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 40 # 8 acting with signature (0; +; [-]; {(2,2,4,4)}) ... and with signature (0; +; [2]; {(4,4)}) ... and with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 40 # 10 acting with signature (0; +; [-]; {(1),(2)}) Group of order 40 # 12 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 80 # 39 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 240 # 189 acting with signature (0; +; [-]; {(2,4,6)}) Number of different groups with symmetric cross-cap number 12 = 7 ........................................................................... Groups with symmetric cross-cap number 13 Group of order 42 # 3 acting with signature (0; +; [14]; {(3)}) Group of order 42 # 4 acting with signature (0; +; [6]; {(7)}) Group of order 52 # 3 acting with signature (0; +; [4]; {(13)}) Group of order 60 # 9 acting with signature (0; +; [15]; {(2)}) Group of order 84 # 8 acting with signature (0; +; [-]; {(2,6,14)}) Group of order 120 # 38 acting with signature (0; +; [-]; {(2,4,15)}) Number of different groups with symmetric cross-cap number 13 = 6 ........................................................................... Groups with symmetric cross-cap number 14 Group of order 16 # 12 acting with signature (0; +; [4]; {(1),(1)}) Group of order 24 # 4 acting with signature (0; +; [4,4]; {(1)}) Group of order 24 # 7 acting with signature (0; +; [4,4]; {(1)}) Group of order 24 # 15 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 32 # 48 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 36 # 11 acting with signature (0; +; [3,3]; {(1)}) ... and with signature (1; -; [3,3]; {-}) Group of order 36 # 12 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) ... and with signature (0; +; [-]; {(1),(3)}) Group of order 48 # 6 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 48 # 14 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 48 # 21 acting with signature (0; +; [-]; {(1),(2)}) Group of order 48 # 24 acting with signature (0; +; [-]; {(1),(2)}) Group of order 48 # 37 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 48 # 43 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 48 # 49 acting with signature (0; +; [-]; {(1),(2)}) Group of order 48 # 51 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 72 # 42 acting with signature (0; +; [2,3]; {(1)}) Group of order 72 # 43 acting with signature (0; +; [-]; {(2,2,3,3)}) ... and with signature (0; +; [2]; {(3,3)}) Group of order 72 # 44 acting with signature (0; +; [3]; {(2,2)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 72 # 46 acting with signature (0; +; [-]; {(2,2,2,6)}) Group of order 96 # 89 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 96 # 115 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 96 # 226 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 144 # 183 acting with signature (0; +; [-]; {(2,2,2,3)}) Group of order 180 # 19 acting with signature (0; +; [3]; {(5)}) Group of order 360 # 121 acting with signature (0; +; [-]; {(2,3,10)}) Number of different groups with symmetric cross-cap number 14 = 25 ........................................................................... Groups with symmetric cross-cap number 15 Group of order 24 # 1 acting with signature (0; +; [3,8]; {(1)}) Group of order 39 # 1 acting with signature (1; -; [3,3]; {-}) Group of order 48 # 15 acting with signature (0; +; [-]; {(2,2,3,8)}) Group of order 78 # 1 acting with signature (0; +; [2,3]; {(1)}) Group of order 1092 # 25 acting with signature (0; +; [-]; {(2,3,7)}) Number of different groups with symmetric cross-cap number 15 = 5 ........................................................................... Groups with symmetric cross-cap number 16 Group of order 28 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 56 # 4 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 56 # 7 acting with signature (0; +; [-]; {(2,2,4,4)}) ... and with signature (0; +; [2]; {(4,4)}) ... and with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 56 # 9 acting with signature (0; +; [-]; {(1),(2)}) Group of order 72 # 16 acting with signature (0; +; [18]; {(2)}) Group of order 112 # 31 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 144 # 109 acting with signature (0; +; [-]; {(2,4,18)}) Number of different groups with symmetric cross-cap number 16 = 7 ........................................................................... Groups with symmetric cross-cap number 17 Group of order 25 # 2 acting with signature (1; -; [5,5]; {-}) Group of order 27 # 2 acting with signature (1; -; [3,9]; {-}) Group of order 27 # 4 acting with signature (1; -; [3,9]; {-}) Group of order 36 # 6 acting with signature (0; +; [3,4]; {(1)}) Group of order 50 # 3 acting with signature (0; +; [10]; {(5)}) ... and with signature (0; +; [2,5]; {(1)}) Group of order 50 # 4 acting with signature (0; +; [2]; {(5,5)}) Group of order 54 # 3 acting with signature (0; +; [6]; {(9)}) Group of order 54 # 4 acting with signature (0; +; [18]; {(3)}) Group of order 54 # 6 acting with signature (0; +; [6]; {(9)}) Group of order 54 # 7 acting with signature (0; +; [2]; {(3,9)}) Group of order 68 # 3 acting with signature (0; +; [4]; {(17)}) Group of order 72 # 23 acting with signature (0; +; [-]; {(2,2,3,4)}) Group of order 72 # 39 acting with signature (0; +; [8]; {(3)}) Group of order 100 # 12 acting with signature (0; +; [4]; {(5)}) Group of order 100 # 13 acting with signature (0; +; [-]; {(2,10,10)}) ... and with signature (0; +; [-]; {(2,2,2,5)}) Group of order 108 # 16 acting with signature (0; +; [-]; {(2,6,18)}) Group of order 200 # 43 acting with signature (0; +; [-]; {(2,4,10)}) Group of order 360 # 118 acting with signature (0; +; [-]; {(3,3,4)}) Group of order 720 # 764 acting with signature (0; +; [-]; {(2,3,8)}) Number of different groups with symmetric cross-cap number 17 = 19 ........................................................................... Groups with symmetric cross-cap number 18 Group of order 24 # 11 acting with signature (0; +; [4,12]; {(1)}) Group of order 32 # 2 acting with signature (0; +; [4,4]; {(1)}) Group of order 32 # 10 acting with signature (0; +; [4,4]; {(1)}) Group of order 32 # 13 acting with signature (0; +; [4,4]; {(1)}) Group of order 32 # 14 acting with signature (0; +; [4,4]; {(1)}) Group of order 32 # 20 acting with signature (0; +; [4,4]; {(1)}) Group of order 32 # 22 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 32 # 25 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 32 # 30 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 32 # 36 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 32 # 37 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 32 # 38 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 32 # 40 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 32 # 44 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 32 # 51 acting with signature (0; +; [-]; {(2,2,2,2,2,2)}) Group of order 48 # 3 acting with signature (0; +; [3,3]; {(1)}) ... and with signature (1; -; [3,3]; {-}) Group of order 48 # 4 acting with signature (0; +; [-]; {(1),(3)}) Group of order 48 # 25 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 48 # 41 acting with signature (0; +; [-]; {(2,2,4,12)}) Group of order 64 # 4 acting with signature (0; +; [-]; {(1),(2)}) Group of order 64 # 8 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 64 # 29 acting with signature (0; +; [-]; {(1),(2)}) Group of order 64 # 30 acting with signature (0; +; [-]; {(1),(2)}) Group of order 64 # 32 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) ... and with signature (0; +; [-]; {(1),(2)}) Group of order 64 # 34 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 64 # 38 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 64 # 41 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 64 # 51 acting with signature (0; +; [-]; {(1),(2)}) Group of order 64 # 53 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 64 # 73 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 64 # 130 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 64 # 147 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 64 # 150 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 64 # 189 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 64 # 202 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 64 # 215 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 64 # 226 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 64 # 250 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 64 # 254 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 64 # 256 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 64 # 257 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 96 # 64 acting with signature (0; +; [-]; {(2,2,3,3)}) ... and with signature (0; +; [2]; {(3,3)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 96 # 72 acting with signature (0; +; [3]; {(2,2)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 96 # 117 acting with signature (0; +; [-]; {(2,2,2,6)}) Group of order 96 # 229 acting with signature (0; +; [3]; {(2,2)}) Group of order 128 # 327 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 128 # 916 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 128 # 922 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 128 # 928 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 128 # 995 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 192 # 956 acting with signature (0; +; [-]; {(2,2,2,3)}) Group of order 192 # 1538 acting with signature (0; +; [-]; {(2,2,2,3)}) Number of different groups with symmetric cross-cap number 18 = 52 ........................................................................... Groups with symmetric cross-cap number 19 Group of order 84 # 10 acting with signature (0; +; [21]; {(2)}) Group of order 168 # 46 acting with signature (0; +; [-]; {(2,4,21)}) Number of different groups with symmetric cross-cap number 19 = 2 ........................................................................... Groups with symmetric cross-cap number 20 Group of order 36 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 48 # 5 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 48 # 17 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 48 # 26 acting with signature (0; +; [-]; {(1),(4)}) Group of order 72 # 5 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 72 # 8 acting with signature (0; +; [-]; {(2,2,4,4)}) ... and with signature (0; +; [2]; {(4,4)}) ... and with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 72 # 10 acting with signature (0; +; [-]; {(1),(2)}) Group of order 72 # 45 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 96 # 121 acting with signature (0; +; [-]; {(2,2,2,8)}) Group of order 144 # 41 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 144 # 186 acting with signature (0; +; [-]; {(2,2,2,4)}) Number of different groups with symmetric cross-cap number 20 = 11 ........................................................................... Groups with symmetric cross-cap number 21 Group of order 57 # 1 acting with signature (1; -; [3,3]; {-}) Group of order 60 # 6 acting with signature (0; +; [12]; {(5)}) Group of order 66 # 1 acting with signature (0; +; [22]; {(3)}) Group of order 66 # 2 acting with signature (0; +; [6]; {(11)}) Group of order 114 # 1 acting with signature (0; +; [2,3]; {(1)}) Group of order 132 # 5 acting with signature (0; +; [-]; {(2,6,22)}) Number of different groups with symmetric cross-cap number 21 = 6 ........................................................................... Groups with symmetric cross-cap number 22 Group of order 32 # 3 acting with signature (0; +; [4,8]; {(1)}) Group of order 32 # 4 acting with signature (0; +; [4,8]; {(1)}) Group of order 32 # 8 acting with signature (0; +; [4,8]; {(1)}) Group of order 32 # 12 acting with signature (0; +; [4,8]; {(1)}) Group of order 32 # 31 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 32 # 50 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 40 # 4 acting with signature (0; +; [4,4]; {(1)}) Group of order 40 # 7 acting with signature (0; +; [4,4]; {(1)}) Group of order 40 # 14 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 48 # 28 acting with signature (0; +; [3,4]; {(1)}) Group of order 48 # 30 acting with signature (0; +; [3,4]; {(1)}) Group of order 48 # 32 acting with signature (0; +; [3,4]; {(1)}) Group of order 60 # 10 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 60 # 11 acting with signature (0; +; [-]; {(1),(3)}) Group of order 64 # 135 acting with signature (0; +; [-]; {(2,2,4,8)}) Group of order 64 # 140 acting with signature (0; +; [-]; {(2,2,4,8)}) Group of order 64 # 174 acting with signature (0; +; [-]; {(2,2,4,8)}) Group of order 64 # 177 acting with signature (0; +; [-]; {(2,2,4,8)}) Group of order 80 # 6 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 80 # 14 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 80 # 21 acting with signature (0; +; [-]; {(1),(2)}) Group of order 80 # 24 acting with signature (0; +; [-]; {(1),(2)}) Group of order 80 # 34 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 80 # 38 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 80 # 44 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 80 # 51 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 96 # 73 acting with signature (0; +; [24]; {(2)}) Group of order 96 # 189 acting with signature (0; +; [-]; {(2,2,3,4)}) Group of order 96 # 192 acting with signature (0; +; [-]; {(2,2,3,4)}) Group of order 96 # 195 acting with signature (0; +; [-]; {(2,2,3,4)}) Group of order 112 # 41 acting with signature (0; +; [14]; {(2)}) Group of order 120 # 42 acting with signature (0; +; [-]; {(2,2,2,6)}) Group of order 160 # 103 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 160 # 129 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 192 # 961 acting with signature (0; +; [-]; {(2,4,24)}) Group of order 240 # 190 acting with signature (0; +; [-]; {(2,2,2,3)}) Number of different groups with symmetric cross-cap number 22 = 36 ........................................................................... Groups with symmetric cross-cap number 23 Group of order 36 # 8 acting with signature (0; +; [3,12]; {(1)}) Group of order 42 # 2 acting with signature (0; +; [3,6]; {(1)}) Group of order 63 # 3 acting with signature (1; -; [3,3]; {-}) Group of order 72 # 33 acting with signature (0; +; [-]; {(2,2,3,12)}) Group of order 126 # 9 acting with signature (0; +; [2,3]; {(1)}) Group of order 504 # 157 acting with signature (0; +; [3]; {(4)}) Group of order 1008 # 881 acting with signature (0; +; [-]; {(2,3,8)}) Number of different groups with symmetric cross-cap number 23 = 7 ........................................................................... Groups with symmetric cross-cap number 24 Group of order 44 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 88 # 4 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 88 # 7 acting with signature (0; +; [-]; {(2,2,4,4)}) ... and with signature (0; +; [2]; {(4,4)}) ... and with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 88 # 9 acting with signature (0; +; [-]; {(1),(2)}) Group of order 176 # 31 acting with signature (0; +; [-]; {(2,2,2,4)}) Number of different groups with symmetric cross-cap number 24 = 5 ........................................................................... Groups with symmetric cross-cap number 25 Group of order 70 # 1 acting with signature (0; +; [14]; {(5)}) Group of order 70 # 2 acting with signature (0; +; [10]; {(7)}) Group of order 78 # 3 acting with signature (0; +; [26]; {(3)}) Group of order 78 # 4 acting with signature (0; +; [6]; {(13)}) Group of order 100 # 3 acting with signature (0; +; [4]; {(25)}) Group of order 108 # 3 acting with signature (0; +; [27]; {(2)}) Group of order 140 # 7 acting with signature (0; +; [-]; {(2,10,14)}) Group of order 156 # 11 acting with signature (0; +; [-]; {(2,6,26)}) Group of order 216 # 21 acting with signature (0; +; [-]; {(2,4,27)}) Number of different groups with symmetric cross-cap number 25 = 9 ........................................................................... Groups with symmetric cross-cap number 26 Group of order 32 # 15 acting with signature (0; +; [8,8]; {(1)}) Group of order 32 # 21 acting with signature (0; +; [4]; {(1),(1)}) Group of order 32 # 23 acting with signature (0; +; [4]; {(1),(1)}) Group of order 32 # 24 acting with signature (0; +; [4]; {(1),(1)}) Group of order 32 # 29 acting with signature (0; +; [4]; {(1),(1)}) Group of order 32 # 41 acting with signature (0; +; [4]; {(1),(1)}) Group of order 32 # 45 acting with signature (0; +; [4]; {(2,2,2,2)}) ... and with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 36 # 14 acting with signature (0; +; [6,6]; {(1)}) ... and with signature (0; +; [3]; {(1),(1)}) Group of order 48 # 8 acting with signature (0; +; [4,4]; {(1)}) Group of order 48 # 11 acting with signature (0; +; [4,4]; {(1)}) Group of order 48 # 12 acting with signature (0; +; [4,4]; {(1)}) Group of order 48 # 13 acting with signature (0; +; [4,4]; {(1)}) Group of order 48 # 19 acting with signature (0; +; [4,4]; {(1)}) Group of order 48 # 35 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 48 # 39 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 48 # 44 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 48 # 45 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 48 # 47 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 6 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) ... and with signature (0; +; [-]; {(1),(4)}) Group of order 64 # 10 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) ... and with signature (0; +; [-]; {(1),(4)}) Group of order 64 # 12 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) ... and with signature (0; +; [-]; {(1),(4)}) Group of order 64 # 31 acting with signature (0; +; [-]; {(1),(4)}) Group of order 64 # 36 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) ... and with signature (0; +; [-]; {(1),(4)}) Group of order 64 # 40 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 64 # 42 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 64 # 153 acting with signature (0; +; [-]; {(2,2,8,8)}) Group of order 64 # 203 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 64 # 211 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 64 # 216 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 64 # 227 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 64 # 242 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 64 # 253 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 64 # 258 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 72 # 21 acting with signature (0; +; [-]; {(1),(3)}) Group of order 72 # 25 acting with signature (0; +; [3,3]; {(1)}) Group of order 72 # 27 acting with signature (0; +; [-]; {(1),(3)}) Group of order 72 # 28 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 72 # 30 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 72 # 49 acting with signature (0; +; [-]; {(2,2,6,6)}) ... and with signature (0; +; [-]; {(2,2,2,2,3)}) Group of order 96 # 7 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 96 # 13 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 96 # 28 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 96 # 32 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 96 # 48 acting with signature (0; +; [-]; {(1),(2)}) Group of order 96 # 49 acting with signature (0; +; [-]; {(1),(2)}) Group of order 96 # 50 acting with signature (0; +; [-]; {(1),(2)}) Group of order 96 # 60 acting with signature (0; +; [-]; {(1),(2)}) Group of order 96 # 91 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 96 # 111 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 96 # 137 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 96 # 144 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 96 # 147 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 96 # 186 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 96 # 196 acting with signature (0; +; [-]; {(1),(2)}) Group of order 96 # 197 acting with signature (0; +; [-]; {(1),(2)}) Group of order 96 # 207 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 96 # 209 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 96 # 211 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 96 # 216 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 128 # 351 acting with signature (0; +; [-]; {(2,2,2,8)}) Group of order 128 # 387 acting with signature (0; +; [-]; {(2,2,2,8)}) Group of order 128 # 924 acting with signature (0; +; [-]; {(2,2,2,8)}) Group of order 128 # 934 acting with signature (0; +; [-]; {(2,2,2,8)}) Group of order 144 # 122 acting with signature (0; +; [3]; {(2,2)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 144 # 125 acting with signature (0; +; [-]; {(2,2,3,3)}) Group of order 144 # 127 acting with signature (0; +; [3]; {(2,2)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 144 # 144 acting with signature (0; +; [-]; {(2,2,2,6)}) Group of order 144 # 154 acting with signature (0; +; [-]; {(2,2,2,6)}) Group of order 160 # 235 acting with signature (0; +; [10]; {(2)}) Group of order 192 # 291 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 192 # 300 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 192 # 306 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 192 # 467 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 192 # 1470 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 192 # 1472 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 288 # 847 acting with signature (0; +; [-]; {(2,2,2,3)}) Group of order 320 # 1636 acting with signature (0; +; [-]; {(2,4,10)}) Number of different groups with symmetric cross-cap number 26 = 77 ........................................................................... Groups with symmetric cross-cap number 27 Group of order 40 # 3 acting with signature (0; +; [4,8]; {(1)}) Group of order 75 # 2 acting with signature (1; -; [3,3]; {-}) Group of order 150 # 5 acting with signature (0; +; [2]; {(3,3)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 150 # 6 acting with signature (0; +; [2,3]; {(1)}) Group of order 300 # 25 acting with signature (0; +; [-]; {(2,2,2,3)}) Number of different groups with symmetric cross-cap number 27 = 5 ........................................................................... Groups with symmetric cross-cap number 28 Group of order 52 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 104 # 5 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 104 # 8 acting with signature (0; +; [-]; {(2,2,4,4)}) ... and with signature (0; +; [2]; {(4,4)}) ... and with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 104 # 10 acting with signature (0; +; [-]; {(1),(2)}) Group of order 104 # 12 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 120 # 43 acting with signature (0; +; [30]; {(2)}) Group of order 208 # 39 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 240 # 197 acting with signature (0; +; [-]; {(2,4,30)}) Number of different groups with symmetric cross-cap number 28 = 8 ........................................................................... Groups with symmetric cross-cap number 29 Group of order 27 # 5 acting with signature (1; -; [3,3,3]; {-}) Group of order 40 # 1 acting with signature (0; +; [5,8]; {(1)}) Group of order 45 # 2 acting with signature (1; -; [3,15]; {-}) Group of order 54 # 10 acting with signature (0; +; [3,6]; {(1)}) Group of order 54 # 12 acting with signature (0; +; [3,6]; {(1)}) Group of order 54 # 13 acting with signature (0; +; [6]; {(3,3)}) ... and with signature (0; +; [2,3]; {(3)}) Group of order 54 # 14 acting with signature (0; +; [2]; {(3,3,3)}) Group of order 80 # 15 acting with signature (0; +; [-]; {(2,2,5,8)}) Group of order 81 # 7 acting with signature (1; -; [3,3]; {-}) Group of order 81 # 9 acting with signature (1; -; [3,3]; {-}) Group of order 90 # 6 acting with signature (0; +; [30]; {(3)}) Group of order 90 # 7 acting with signature (0; +; [6]; {(15)}) Group of order 90 # 9 acting with signature (0; +; [2]; {(3,15)}) Group of order 108 # 28 acting with signature (0; +; [-]; {(2,2,3,6)}) Group of order 108 # 36 acting with signature (0; +; [12]; {(3)}) Group of order 108 # 39 acting with signature (0; +; [-]; {(2,2,3,6)}) Group of order 108 # 40 acting with signature (0; +; [-]; {(6,6,6)}) ... and with signature (0; +; [-]; {(2,3,2,6)}) Group of order 116 # 3 acting with signature (0; +; [4]; {(29)}) Group of order 162 # 10 acting with signature (0; +; [6]; {(3)}) Group of order 162 # 11 acting with signature (0; +; [6]; {(3)}) Group of order 162 # 14 acting with signature (0; +; [2,3]; {(1)}) Group of order 162 # 15 acting with signature (0; +; [2,3]; {(1)}) Group of order 162 # 19 acting with signature (0; +; [2]; {(3,3)}) Group of order 162 # 21 acting with signature (0; +; [2]; {(3,3)}) Group of order 180 # 29 acting with signature (0; +; [-]; {(2,6,30)}) Group of order 216 # 159 acting with signature (0; +; [-]; {(2,6,12)}) Group of order 324 # 39 acting with signature (0; +; [-]; {(2,6,6)}) Group of order 324 # 41 acting with signature (0; +; [-]; {(2,2,2,3)}) Group of order 324 # 160 acting with signature (0; +; [3]; {(6)}) Group of order 648 # 703 acting with signature (0; +; [-]; {(2,3,12)}) Number of different groups with symmetric cross-cap number 29 = 30 ........................................................................... Groups with symmetric cross-cap number 30 Group of order 40 # 11 acting with signature (0; +; [4,20]; {(1)}) Group of order 48 # 16 acting with signature (0; +; [4,6]; {(1)}) Group of order 56 # 3 acting with signature (0; +; [4,4]; {(1)}) Group of order 56 # 6 acting with signature (0; +; [4,4]; {(1)}) Group of order 56 # 13 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 80 # 42 acting with signature (0; +; [-]; {(2,2,4,20)}) Group of order 84 # 7 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 84 # 11 acting with signature (0; +; [3,3]; {(1)}) ... and with signature (1; -; [3,3]; {-}) Group of order 84 # 12 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 84 # 13 acting with signature (0; +; [-]; {(1),(3)}) Group of order 96 # 118 acting with signature (0; +; [-]; {(2,2,4,6)}) Group of order 96 # 204 acting with signature (0; +; [3]; {(2,4)}) Group of order 112 # 5 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 112 # 13 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 112 # 20 acting with signature (0; +; [-]; {(1),(2)}) Group of order 112 # 23 acting with signature (0; +; [-]; {(1),(2)}) Group of order 112 # 30 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 112 # 36 acting with signature (0; +; [-]; {(2,2,4,4)}) Group of order 112 # 42 acting with signature (0; +; [-]; {(2,2,2,2,2)}) Group of order 160 # 199 acting with signature (0; +; [5]; {(4)}) Group of order 168 # 49 acting with signature (0; +; [3]; {(2,2)}) ... and with signature (0; +; [2,3]; {(1)}) Group of order 168 # 50 acting with signature (0; +; [-]; {(2,2,2,6)}) Group of order 192 # 1494 acting with signature (0; +; [-]; {(3,4,8)}) Group of order 224 # 77 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 224 # 103 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 320 # 1582 acting with signature (0; +; [-]; {(2,5,8)}) Group of order 336 # 209 acting with signature (0; +; [-]; {(3,4,4)}) Group of order 672 # 1254 acting with signature (0; +; [-]; {(2,4,6)}) Number of different groups with symmetric cross-cap number 30 = 28 ........................................................................... Groups with symmetric cross-cap number 31 Group of order 48 # 1 acting with signature (0; +; [3,16]; {(1)}) Group of order 96 # 33 acting with signature (0; +; [-]; {(2,2,3,16)}) Group of order 132 # 6 acting with signature (0; +; [33]; {(2)}) Group of order 264 # 32 acting with signature (0; +; [-]; {(2,4,33)}) Number of different groups with symmetric cross-cap number 31 = 4 ........................................................................... Groups with symmetric cross-cap number 32 Group of order 48 # 18 acting with signature (0; +; [4,8]; {(1)}) Group of order 60 # 3 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 60 # 7 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 80 # 4 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 80 # 5 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 80 # 17 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 80 # 25 acting with signature (0; +; [-]; {(1),(4)}) Group of order 80 # 26 acting with signature (0; +; [-]; {(1),(4)}) Group of order 80 # 28 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 80 # 29 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 96 # 123 acting with signature (0; +; [-]; {(2,2,4,8)}) Group of order 120 # 27 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 120 # 30 acting with signature (0; +; [-]; {(2,2,4,4)}) ... and with signature (0; +; [2]; {(4,4)}) ... and with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 120 # 32 acting with signature (0; +; [-]; {(1),(2)}) Group of order 120 # 36 acting with signature (0; +; [4]; {(2,2)}) ... and with signature (0; +; [2,4]; {(1)}) Group of order 160 # 131 acting with signature (0; +; [-]; {(2,2,2,8)}) Group of order 160 # 135 acting with signature (0; +; [-]; {(2,2,2,8)}) Group of order 240 # 179 acting with signature (0; +; [-]; {(2,2,2,4)}) Group of order 360 # 119 acting with signature (0; +; [6]; {(2)}) Group of order 360 # 120 acting with signature (0; +; [4]; {(3)}) Group of order 720 # 767 acting with signature (0; +; [-]; {(2,4,6)}) Number of different groups with symmetric cross-cap number 32 = 21 ........................................................................... Groups with symmetric cross-cap number 33 Group of order 90 # 1 acting with signature (0; +; [10]; {(9)}) Group of order 90 # 2 acting with signature (0; +; [18]; {(5)}) Group of order 93 # 1 acting with signature (1; -; [3,3]; {-}) Group of order 102 # 1 acting with signature (0; +; [34]; {(3)}) Group of order 102 # 2 acting with signature (0; +; [6]; {(17)}) Group of order 180 # 7 acting with signature (0; +; [-]; {(2,10,18)}) Group of order 186 # 1 acting with signature (0; +; [2,3]; {(1)}) Group of order 204 # 7 acting with signature (0; +; [-]; {(2,6,34)}) Number of different groups with symmetric cross-cap number 33 = 8 ........................................................................... Groups with symmetric cross-cap number 34 Group of order 32 # 33 acting with signature (0; +; [4,4]; {(2,2)}) ... and with signature (0; +; [2,4,4]; {(1)}) ... and with signature (0; +; [4]; {(1),(2)}) Group of order 32 # 47 acting with signature (0; +; [-]; {(1),(1),(1)}) Group of order 48 # 9 acting with signature (0; +; [3]; {(1),(1)}) Group of order 48 # 20 acting with signature (0; +; [4,12]; {(1)}) Group of order 48 # 22 acting with signature (0; +; [4,12]; {(1)}) Group of order 48 # 27 acting with signature (0; +; [4,12]; {(1)}) Group of order 64 # 9 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 18 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 20 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 21 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 23 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 33 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 35 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 39 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 47 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 48 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 54 acting with signature (0; +; [4,4]; {(1)}) Group of order 64 # 60 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) Group of order 64 # 67 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 87 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 88 acting with signature (0; +; [-]; {(2,2),(1)}) Group of order 64 # 89 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 90 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 92 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 94 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 95 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 97 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 98 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 99 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) Group of order 64 # 101 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 102 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 115 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 116 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 118 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 123 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 124 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 125 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 129 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 131 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) Group of order 64 # 136 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) Group of order 64 # 139 acting with signature (0; +; [4]; {(2,2,2)}) Group of order 64 # 144 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 146 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 152 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 161 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 162 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 183 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 184 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 185 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 187 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 191 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 64 # 206 acting with signature (0; +; [-]; {(2,2,2,4,4)}) ... and with signature (0; +; [-]; {(2,2,4,2,4)}) Group of order 64 # 219 acting with signature (0; +; [-]; {(2,2,2,4,4)}) ... and with signature (0; +; [-]; {(2,2,4,2,4)}) Group of order 64 # 241 acting with signature (0; +; [-]; {(2,2,2,4,4)}) ... and with signature (0; +; [-]; {(2,2,4,2,4)}) Group of order 64 # 261 acting with signature (0; +; [-]; {(2,2,2,2,2,2)}) Group of order 64 # 263 acting with signature (0; +; [-]; {(2,2,2,2,2,2)}) Group of order 64 # 264 acting with signature (0; +; [-]; {(2,2,2,2,2,2)}) Group of order 64 # 266 acting with signature (0; +; [-]; {(2,2,2,2,2,2)}) Group of order 96 # 3 acting with signature (0; +; [3,3]; {(1)}) Group of order 96 # 4 acting with signature (0; +; [-]; {(1),(3)}) Group of order 96 # 61 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 96 # 74 acting with signature (0; +; [24]; {(4)}) Group of order 96 # 81 acting with signature (0; +; [-]; {(2,2,4,12)}) Group of order 96 # 102 acting with signature (0; +; [-]; {(2,2,4,12)}) Group of order 96 # 126 acting with signature (0; +; [-]; {(2,2,4,12)}) Group of order 96 # 138 acting with signature (0; +; [-]; {(2,2,2,2,3)}) Group of order 96 # 190 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) ... and with signature (0; +; [-]; {(1),(3)}) Group of order 96 # 200 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 96 # 202 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 96 # 203 acting with signature (0; +; [3,3]; {(1)}) Number of different groups with symmetric cross-cap number 34 = 70 ........................................................................... Groups with symmetric cross-cap number 35 Group of order 54 # 9 acting with signature (0; +; [3,18]; {(1)}) Group of order 54 # 11 acting with signature (0; +; [3,18]; {(1)}) Group of order 55 # 1 acting with signature (1; -; [5,5]; {-}) Group of order 108 # 27 acting with signature (0; +; [-]; {(2,2,3,18)}) Group of order 110 # 1 acting with signature (0; +; [2,5]; {(1)}) Number of different groups with symmetric cross-cap number 35 = 5 ........................................................................... Groups with symmetric cross-cap number 36 Group of order 48 # 10 acting with signature (0; +; [6,8]; {(1)}) Group of order 68 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 96 # 156 acting with signature (0; +; [-]; {(2,2,6,8)}) Group of order 120 # 37 acting with signature (0; +; [20]; {(3)}) Group of order 120 # 39 acting with signature (0; +; [6]; {(10)}) Number of different groups with symmetric cross-cap number 36 = 5 ........................................................................... Groups with symmetric cross-cap number 37 Group of order 49 # 2 acting with signature (1; -; [7,7]; {-}) Group of order 50 # 5 acting with signature (0; +; [5,10]; {(1)}) Group of order 84 # 1 acting with signature (0; +; [3,4]; {(1)}) Group of order 98 # 3 acting with signature (0; +; [14]; {(7)}) ... and with signature (0; +; [2,7]; {(1)}) Group of order 98 # 4 acting with signature (0; +; [2]; {(7,7)}) Group of order 100 # 9 acting with signature (0; +; [20]; {(5)}) Group of order 100 # 15 acting with signature (0; +; [-]; {(2,2,5,10)}) Group of order 114 # 3 acting with signature (0; +; [38]; {(3)}) Group of order 114 # 4 acting with signature (0; +; [6]; {(19)}) Number of different groups with symmetric cross-cap number 37 = 9 ........................................................................... Groups with symmetric cross-cap number 38 Group of order 48 # 34 acting with signature (0; +; [4]; {(1),(1)}) Group of order 48 # 40 acting with signature (0; +; [4]; {(1),(1)}) Group of order 48 # 42 acting with signature (0; +; [4]; {(1),(1)}) Group of order 48 # 46 acting with signature (0; +; [4]; {(1),(1)}) Group of order 48 # 52 acting with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 72 # 4 acting with signature (0; +; [4,4]; {(1)}) Group of order 72 # 7 acting with signature (0; +; [4,4]; {(1)}) Group of order 72 # 18 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 72 # 41 acting with signature (0; +; [4,4]; {(1)}) Group of order 72 # 47 acting with signature (0; +; [3,6]; {(1)}) Group of order 72 # 48 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 12 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 96 # 16 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 96 # 27 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 96 # 30 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 96 # 52 acting with signature (0; +; [-]; {(1),(4)}) Group of order 96 # 54 acting with signature (0; +; [-]; {(1),(4)}) Group of order 96 # 201 acting with signature (0; +; [-]; {(1),(4)}) Group of order 96 # 208 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 96 # 213 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 96 # 215 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 96 # 219 acting with signature (0; +; [-]; {(2,2,2,2,4)}) Group of order 108 # 22 acting with signature (0; +; [3,3]; {(1)}) ... and with signature (1; -; [3,3]; {-}) Group of order 108 # 23 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 108 # 24 acting with signature (0; +; [-]; {(1),(3)}) Group of order 108 # 25 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) ... and with signature (0; +; [-]; {(1),(3)}) Group of order 108 # 26 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 108 # 38 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) ... and with signature (0; +; [-]; {(1),(3)}) Number of different groups with symmetric cross-cap number 38 = 28 ........................................................................... Groups with symmetric cross-cap number 39 Group of order 60 # 1 acting with signature (0; +; [3,20]; {(1)}) Group of order 111 # 1 acting with signature (1; -; [3,3]; {-}) Group of order 120 # 12 acting with signature (0; +; [-]; {(2,2,3,20)}) Number of different groups with symmetric cross-cap number 39 = 3 ........................................................................... Groups with symmetric cross-cap number 40 Group of order 76 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Number of different groups with symmetric cross-cap number 40 = 1 ........................................................................... Groups with symmetric cross-cap number 41 Group of order 36 # 7 acting with signature (0; +; [3,3,4]; {(1)}) Group of order 63 # 4 acting with signature (1; -; [3,21]; {-}) Group of order 72 # 12 acting with signature (0; +; [3,8]; {(1)}) Group of order 72 # 19 acting with signature (0; +; [3,8]; {(1)}) Group of order 72 # 22 acting with signature (0; +; [-]; {(2,2,3,4,3)}) ... and with signature (0; +; [3]; {(2,2,4)}) Group of order 72 # 35 acting with signature (0; +; [-]; {(2,2,3,3,4)}) Group of order 78 # 2 acting with signature (0; +; [3,6]; {(1)}) Group of order 110 # 3 acting with signature (0; +; [22]; {(5)}) Group of order 110 # 4 acting with signature (0; +; [10]; {(11)}) Group of order 117 # 3 acting with signature (1; -; [3,3]; {-}) Group of order 126 # 12 acting with signature (0; +; [42]; {(3)}) Group of order 126 # 13 acting with signature (0; +; [6]; {(21)}) Group of order 126 # 15 acting with signature (0; +; [2]; {(3,21)}) Number of different groups with symmetric cross-cap number 41 = 13 ........................................................................... Groups with symmetric cross-cap number 42 Group of order 32 # 26 acting with signature (0; +; [4,4,4]; {(1)}) Group of order 32 # 32 acting with signature (0; +; [4,4,4]; {(1)}) Group of order 32 # 35 acting with signature (0; +; [4,4,4]; {(1)}) Group of order 56 # 10 acting with signature (0; +; [4,28]; {(1)}) Group of order 60 # 2 acting with signature (0; +; [4,12]; {(1)}) ... and with signature (1; -; [4,12]; {-}) Group of order 64 # 5 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 7 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 11 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 13 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 14 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 17 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 24 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 25 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 37 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 43 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 46 acting with signature (0; +; [4,8]; {(1)}) Group of order 64 # 71 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 75 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 91 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 133 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 137 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 141 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 149 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 163 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 167 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 171 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 173 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 64 # 213 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 64 # 218 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 64 # 221 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 64 # 228 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 64 # 229 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 64 # 231 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 64 # 234 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 64 # 236 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 64 # 243 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 64 # 259 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 80 # 8 acting with signature (0; +; [4,4]; {(1)}) Group of order 80 # 11 acting with signature (0; +; [4,4]; {(1)}) Group of order 80 # 12 acting with signature (0; +; [4,4]; {(1)}) Group of order 80 # 13 acting with signature (0; +; [4,4]; {(1)}) Group of order 80 # 19 acting with signature (0; +; [4,4]; {(1)}) Group of order 80 # 30 acting with signature (0; +; [4,4]; {(1)}) Group of order 80 # 31 acting with signature (0; +; [4,4]; {(1)}) Group of order 80 # 36 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 80 # 40 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 80 # 45 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 80 # 46 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 80 # 48 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 80 # 50 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 5 acting with signature (0; +; [-]; {(1),(6)}) Group of order 96 # 35 acting with signature (0; +; [12]; {(2,2)}) ... and with signature (0; +; [2,12]; {(1)}) Group of order 96 # 62 acting with signature (0; +; [12]; {(2,2)}) ... and with signature (0; +; [2,12]; {(1)}) Group of order 96 # 66 acting with signature (0; +; [3,4]; {(1)}) Group of order 96 # 67 acting with signature (0; +; [3,4]; {(1)}) Group of order 96 # 68 acting with signature (0; +; [3,4]; {(1)}) Group of order 96 # 69 acting with signature (0; +; [3,4]; {(1)}) Group of order 96 # 71 acting with signature (0; +; [3,4]; {(1)}) Group of order 100 # 14 acting with signature (0; +; [10]; {(2,2)}) ... and with signature (0; +; [2,10]; {(1)}) ... and with signature (0; +; [-]; {(1),(5)}) Group of order 112 # 34 acting with signature (0; +; [-]; {(2,2,4,28)}) Group of order 120 # 5 acting with signature (0; +; [3,3]; {(1)}) Group of order 120 # 13 acting with signature (0; +; [-]; {(2,2,4,12)}) ... and with signature (0; +; [2]; {(4,12)}) Group of order 120 # 18 acting with signature (0; +; [6]; {(2,2)}) ... and with signature (0; +; [2,6]; {(1)}) Group of order 120 # 22 acting with signature (0; +; [-]; {(1),(3)}) Number of different groups with symmetric cross-cap number 42 = 64 ........................................................................... Groups with symmetric cross-cap number 43 Group of order 56 # 1 acting with signature (0; +; [7,8]; {(1)}) Group of order 112 # 14 acting with signature (0; +; [-]; {(2,2,7,8)}) Number of different groups with symmetric cross-cap number 43 = 2 ........................................................................... Groups with symmetric cross-cap number 44 Group of order 72 # 20 acting with signature (0; +; [4,6]; {(1)}) Group of order 72 # 29 acting with signature (0; +; [4,6]; {(1)}) Group of order 72 # 32 acting with signature (0; +; [2,4]; {(3)}) ... and with signature (0; +; [4]; {(2,2,3)}) Group of order 84 # 5 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 112 # 3 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 112 # 4 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 112 # 16 acting with signature (0; +; [8]; {(2,2)}) ... and with signature (0; +; [2,8]; {(1)}) Group of order 112 # 24 acting with signature (0; +; [-]; {(1),(4)}) Group of order 112 # 25 acting with signature (0; +; [-]; {(1),(4)}) Number of different groups with symmetric cross-cap number 44 = 9 ........................................................................... Groups with symmetric cross-cap number 46 Group of order 64 # 26 acting with signature (0; +; [4,16]; {(1)}) Group of order 64 # 27 acting with signature (0; +; [4,16]; {(1)}) Group of order 64 # 28 acting with signature (0; +; [4,16]; {(1)}) Group of order 64 # 44 acting with signature (0; +; [4,16]; {(1)}) Group of order 64 # 45 acting with signature (0; +; [4,16]; {(1)}) Group of order 64 # 176 acting with signature (0; +; [4]; {(2,2,8)}) Group of order 88 # 3 acting with signature (0; +; [4,4]; {(1)}) Group of order 88 # 6 acting with signature (0; +; [4,4]; {(1)}) Group of order 88 # 12 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 139 acting with signature (0; +; [-]; {(2,2,2,3,4)}) Group of order 96 # 160 acting with signature (0; +; [-]; {(2,2,2,3,4)}) Number of different groups with symmetric cross-cap number 46 = 11 ........................................................................... Groups with symmetric cross-cap number 47 Group of order 72 # 14 acting with signature (0; +; [3,24]; {(1)}) Group of order 81 # 3 acting with signature (1; -; [3,9]; {-}) Group of order 81 # 8 acting with signature (1; -; [3,9]; {-}) Group of order 108 # 8 acting with signature (0; +; [3,4]; {(1)}) Group of order 108 # 37 acting with signature (0; +; [3,4]; {(1)}) Number of different groups with symmetric cross-cap number 47 = 5 ........................................................................... Groups with symmetric cross-cap number 48 Group of order 72 # 3 acting with signature (0; +; [4,9]; {(1)}) ... and with signature (1; -; [4,9]; {-}) Group of order 92 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Number of different groups with symmetric cross-cap number 48 = 2 ........................................................................... Groups with symmetric cross-cap number 49 Group of order 63 # 1 acting with signature (1; -; [7,9]; {-}) Group of order 126 # 1 acting with signature (0; +; [18]; {(7)}) Group of order 126 # 3 acting with signature (0; +; [14]; {(9)}) Group of order 126 # 4 acting with signature (0; +; [18]; {(7)}) Number of different groups with symmetric cross-cap number 49 = 4 ........................................................................... Groups with symmetric cross-cap number 50 Group of order 64 # 2 acting with signature (0; +; [8,8]; {(1)}) Group of order 64 # 3 acting with signature (0; +; [8,8]; {(1)}) Group of order 64 # 15 acting with signature (0; +; [8,8]; {(1)}) Group of order 64 # 16 acting with signature (0; +; [8,8]; {(1)}) Group of order 64 # 19 acting with signature (0; +; [8,8]; {(1)}) Group of order 64 # 22 acting with signature (0; +; [8,8]; {(1)}) Group of order 64 # 49 acting with signature (0; +; [8,8]; {(1)}) Group of order 64 # 56 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 58 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 61 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 62 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 66 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 83 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 84 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 86 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 93 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 96 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 100 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 103 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 105 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 106 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 107 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 108 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 109 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 117 acting with signature (0; +; [8]; {(2,2,4)}) ... and with signature (0; +; [-]; {(2,2),(2)}) ... and with signature (0; +; [-]; {(4,4),(1)}) ... and with signature (0; +; [2]; {(1),(2)}) Group of order 64 # 119 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 121 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 132 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 142 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 148 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 154 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 155 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 157 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 159 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 164 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 165 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 188 acting with signature (0; +; [4]; {(1),(1)}) Group of order 64 # 193 acting with signature (0; +; [4]; {(2,2,2,2)}) ... and with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 196 acting with signature (0; +; [4]; {(2,2,2,2)}) ... and with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 199 acting with signature (0; +; [4]; {(2,2,2,2)}) ... and with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 205 acting with signature (0; +; [4]; {(2,2,2,2)}) ... and with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 207 acting with signature (0; +; [4]; {(2,2,2,2)}) Group of order 64 # 232 acting with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 246 acting with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 247 acting with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 248 acting with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 249 acting with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 251 acting with signature (0; +; [4]; {(2,2,2,2)}) ... and with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 255 acting with signature (0; +; [4]; {(2,2,2,2)}) ... and with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 64 # 265 acting with signature (0; +; [-]; {(2,2,2,2,4,4)}) ... and with signature (0; +; [-]; {(2,2,4,2,2,4)}) Group of order 64 # 267 acting with signature (0; +; [-]; {(2,2,2,2,2,2,2)}) Group of order 72 # 24 acting with signature (0; +; [4,12]; {(1)}) Group of order 72 # 26 acting with signature (0; +; [4,12]; {(1)}) Group of order 72 # 36 acting with signature (0; +; [3]; {(1),(1)}) Group of order 72 # 37 acting with signature (0; +; [6,6]; {(1)}) Group of order 96 # 8 acting with signature (0; +; [4,4]; {(1)}) Group of order 96 # 23 acting with signature (0; +; [4,4]; {(1)}) Group of order 96 # 24 acting with signature (0; +; [4,4]; {(1)}) Group of order 96 # 25 acting with signature (0; +; [4,4]; {(1)}) Group of order 96 # 38 acting with signature (0; +; [4,4]; {(1)}) Group of order 96 # 41 acting with signature (0; +; [4,4]; {(1)}) Group of order 96 # 80 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 87 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 88 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 93 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 100 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 101 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 106 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 107 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 108 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 109 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 113 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 114 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 116 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 120 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 125 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 134 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 135 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 136 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 146 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 148 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 149 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 157 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 162 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 165 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 167 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 168 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 170 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 176 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 177 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 178 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 179 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 180 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 182 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 183 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 184 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 185 acting with signature (0; +; [4,4]; {(1)}) Group of order 96 # 194 acting with signature (0; +; [4,4]; {(1)}) Group of order 96 # 228 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 96 # 230 acting with signature (0; +; [-]; {(2,2,2,2,2,2)}) Group of order 120 # 17 acting with signature (0; +; [-]; {(1),(5)}) Group of order 120 # 23 acting with signature (0; +; [10]; {(2,2)}) ... and with signature (0; +; [2,10]; {(1)}) Group of order 120 # 40 acting with signature (0; +; [-]; {(1),(5)}) Number of different groups with symmetric cross-cap number 50 = 103 ........................................................................... Groups with symmetric cross-cap number 51 Group of order 84 # 2 acting with signature (0; +; [3,12]; {(1)}) Number of different groups with symmetric cross-cap number 51 = 1 ........................................................................... Groups with symmetric cross-cap number 52 Group of order 80 # 16 acting with signature (0; +; [4,8]; {(1)}) Group of order 80 # 18 acting with signature (0; +; [4,8]; {(1)}) Group of order 80 # 32 acting with signature (0; +; [4,8]; {(1)}) Group of order 80 # 33 acting with signature (0; +; [4,8]; {(1)}) Group of order 100 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 100 # 10 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Group of order 120 # 10 acting with signature (0; +; [12]; {(2,2)}) ... and with signature (0; +; [2,12]; {(1)}) Group of order 120 # 20 acting with signature (0; +; [12]; {(2,2)}) ... and with signature (0; +; [2,12]; {(1)}) Group of order 120 # 25 acting with signature (0; +; [-]; {(1),(6)}) Number of different groups with symmetric cross-cap number 52 = 9 ........................................................................... Groups with symmetric cross-cap number 53 Group of order 81 # 5 acting with signature (1; -; [3,27]; {-}) Group of order 81 # 6 acting with signature (1; -; [3,27]; {-}) Group of order 90 # 8 acting with signature (0; +; [10]; {(3,3)}) Number of different groups with symmetric cross-cap number 53 = 3 ........................................................................... Groups with symmetric cross-cap number 54 Group of order 72 # 11 acting with signature (0; +; [4,36]; {(1)}) Group of order 96 # 65 acting with signature (0; +; [3,8]; {(1)}) Group of order 104 # 4 acting with signature (0; +; [4,4]; {(1)}) Group of order 104 # 7 acting with signature (0; +; [4,4]; {(1)}) Group of order 104 # 14 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Number of different groups with symmetric cross-cap number 54 = 5 ........................................................................... Groups with symmetric cross-cap number 55 Group of order 84 # 3 acting with signature (0; +; [3,28]; {(1)}) Number of different groups with symmetric cross-cap number 55 = 1 ........................................................................... Groups with symmetric cross-cap number 56 Group of order 72 # 1 acting with signature (0; +; [8,8]; {(1)}) ... and with signature (1; -; [8,8]; {-}) Group of order 108 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Number of different groups with symmetric cross-cap number 56 = 2 ........................................................................... Groups with symmetric cross-cap number 57 Group of order 75 # 3 acting with signature (1; -; [5,15]; {-}) Group of order 100 # 6 acting with signature (0; +; [4,5]; {(1)}) Group of order 100 # 11 acting with signature (0; +; [4]; {(5,5)}) Number of different groups with symmetric cross-cap number 57 = 3 ........................................................................... Groups with symmetric cross-cap number 58 Group of order 64 # 85 acting with signature (0; +; [8]; {(1),(1)}) Group of order 64 # 104 acting with signature (0; +; [8]; {(1),(1)}) Group of order 64 # 110 acting with signature (0; +; [8]; {(1),(1)}) Group of order 64 # 111 acting with signature (0; +; [8]; {(1),(1)}) Group of order 64 # 112 acting with signature (0; +; [8]; {(1),(1)}) Group of order 64 # 113 acting with signature (0; +; [8]; {(1),(1)}) Group of order 64 # 198 acting with signature (0; +; [-]; {(2,2,4),(1)}) Group of order 64 # 201 acting with signature (0; +; [-]; {(2,2,4),(1)}) Group of order 64 # 210 acting with signature (0; +; [-]; {(2,2,4),(1)}) Group of order 64 # 217 acting with signature (0; +; [-]; {(2,2,4),(1)}) Group of order 64 # 220 acting with signature (0; +; [-]; {(2,2,4),(1)}) Group of order 64 # 223 acting with signature (0; +; [-]; {(2,2,4),(1)}) Group of order 64 # 240 acting with signature (0; +; [-]; {(2,2,4),(1)}) Group of order 80 # 20 acting with signature (0; +; [4,20]; {(1)}) Group of order 80 # 22 acting with signature (0; +; [4,20]; {(1)}) Group of order 80 # 27 acting with signature (0; +; [4,20]; {(1)}) Group of order 84 # 4 acting with signature (0; +; [4,12]; {(1)}) ... and with signature (1; -; [4,12]; {-}) Group of order 84 # 9 acting with signature (0; +; [6,6]; {(1)}) ... and with signature (0; +; [3]; {(1),(1)}) Group of order 96 # 34 acting with signature (0; +; [4,6]; {(1)}) Group of order 96 # 39 acting with signature (0; +; [4,6]; {(1)}) Group of order 96 # 44 acting with signature (0; +; [4,6]; {(1)}) Group of order 96 # 119 acting with signature (0; +; [-]; {(2,2,4,3,4)}) Group of order 96 # 145 acting with signature (0; +; [-]; {(2,2,4,3,4)}) ... and with signature (0; +; [-]; {(2,2,3,4,4)}) Group of order 96 # 188 acting with signature (0; +; [4,6]; {(1)}) Group of order 96 # 198 acting with signature (0; +; [4,6]; {(1)}) Group of order 112 # 7 acting with signature (0; +; [4,4]; {(1)}) Group of order 112 # 10 acting with signature (0; +; [4,4]; {(1)}) Group of order 112 # 11 acting with signature (0; +; [4,4]; {(1)}) Group of order 112 # 12 acting with signature (0; +; [4,4]; {(1)}) Group of order 112 # 18 acting with signature (0; +; [4,4]; {(1)}) Group of order 112 # 28 acting with signature (0; +; [4]; {(2,2,2)}) ... and with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 112 # 32 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 112 # 37 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 112 # 38 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 112 # 40 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Number of different groups with symmetric cross-cap number 58 = 35 ........................................................................... Groups with symmetric cross-cap number 59 Group of order 90 # 10 acting with signature (0; +; [3,30]; {(1)}) Group of order 114 # 2 acting with signature (0; +; [3,6]; {(1)}) Number of different groups with symmetric cross-cap number 59 = 2 ........................................................................... Groups with symmetric cross-cap number 60 Group of order 116 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Number of different groups with symmetric cross-cap number 60 = 1 ........................................................................... Groups with symmetric cross-cap number 61 Group of order 80 # 1 acting with signature (0; +; [5,16]; {(1)}) Group of order 80 # 3 acting with signature (0; +; [5,16]; {(1)}) Number of different groups with symmetric cross-cap number 61 = 2 ........................................................................... Groups with symmetric cross-cap number 62 Group of order 72 # 38 acting with signature (0; +; [12,12]; {(1)}) Group of order 72 # 50 acting with signature (0; +; [6]; {(1),(1)}) Group of order 80 # 9 acting with signature (0; +; [8,8]; {(1)}) Group of order 80 # 10 acting with signature (0; +; [8,8]; {(1)}) Group of order 80 # 35 acting with signature (0; +; [4]; {(1),(1)}) Group of order 80 # 41 acting with signature (0; +; [4]; {(1),(1)}) Group of order 80 # 43 acting with signature (0; +; [4]; {(1),(1)}) Group of order 80 # 47 acting with signature (0; +; [4]; {(1),(1)}) Group of order 80 # 52 acting with signature (0; +; [-]; {(2,2,2),(1)}) Group of order 90 # 5 acting with signature (0; +; [6,6]; {(1)}) ... and with signature (1; -; [6,6]; {-}) Group of order 96 # 14 acting with signature (0; +; [4,8]; {(1)}) Group of order 96 # 15 acting with signature (0; +; [4,8]; {(1)}) Group of order 96 # 17 acting with signature (0; +; [4,8]; {(1)}) Group of order 96 # 20 acting with signature (0; +; [4,8]; {(1)}) Group of order 96 # 21 acting with signature (0; +; [4,8]; {(1)}) Group of order 96 # 22 acting with signature (0; +; [4,8]; {(1)}) Group of order 96 # 31 acting with signature (0; +; [4,8]; {(1)}) Group of order 96 # 42 acting with signature (0; +; [4,8]; {(1)}) Group of order 96 # 82 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 96 # 90 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 96 # 92 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 96 # 122 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 96 # 154 acting with signature (0; +; [4]; {(2,2,4)}) Group of order 96 # 191 acting with signature (0; +; [4,8]; {(1)}) Group of order 96 # 210 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 96 # 214 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 96 # 217 acting with signature (0; +; [-]; {(2,2,4,4,4)}) Group of order 108 # 18 acting with signature (0; +; [3,9]; {(1)}) ... and with signature (1; -; [3,9]; {-}) Group of order 108 # 19 acting with signature (0; +; [3,9]; {(1)}) ... and with signature (1; -; [3,9]; {-}) Group of order 120 # 26 acting with signature (0; +; [4,4]; {(1)}) Group of order 120 # 29 acting with signature (0; +; [4,4]; {(1)}) Group of order 120 # 41 acting with signature (0; +; [4,4]; {(1)}) Group of order 120 # 44 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 120 # 45 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Group of order 120 # 47 acting with signature (0; +; [-]; {(2,2),(1)}) ... and with signature (0; +; [2]; {(1),(1)}) Number of different groups with symmetric cross-cap number 62 = 35 ........................................................................... Groups with symmetric cross-cap number 63 Group of order 96 # 1 acting with signature (0; +; [3,32]; {(1)}) Number of different groups with symmetric cross-cap number 63 = 1 ........................................................................... Groups with symmetric cross-cap number 64 Group of order 124 # 1 acting with signature (0; +; [4,4]; {(1)}) ... and with signature (1; -; [4,4]; {-}) Number of different groups with symmetric cross-cap number 64 = 1 ........................................................................... Groups with symmetric cross-cap number 65 Group of order 54 # 15 acting with signature (0; +; [3,3,6]; {(1)}) Group of order 81 # 2 acting with signature (1; -; [9,9]; {-}) Group of order 81 # 4 acting with signature (1; -; [9,9]; {-}) Group of order 81 # 10 acting with signature (1; -; [9,9]; {-}) Group of order 99 # 2 acting with signature (1; -; [3,33]; {-}) Group of order 105 # 1 acting with signature (1; -; [3,15]; {-}) Group of order 108 # 13 acting with signature (0; +; [3,12]; {(1)}) Group of order 108 # 32 acting with signature (0; +; [3,12]; {(1)}) Group of order 108 # 43 acting with signature (0; +; [3]; {(2,2,6)}) Group of order 108 # 44 acting with signature (0; +; [-]; {(2,2,3,6,3)}) ... and with signature (0; +; [-]; {(2,2,3,3,6)}) Group of order 126 # 7 acting with signature (0; +; [3,6]; {(1)}) Group of order 126 # 8 acting with signature (0; +; [3,6]; {(1)}) Group of order 126 # 10 acting with signature (0; +; [3,6]; {(1)}) Number of different groups with symmetric cross-cap number 65 = 13 ...........................................................................