# An Atlas of Polytopes for Small Almost Simple Groups

by Laurence Vauthier and Dimitri Leemans

This Atlas contains all regular polytopes whose automorphism group is an almost simple group G such that S <= G <= Aut(S) and S is a simple group of order less than 1 million appearing in the Atlas of Finite Groups by Conway et al.
By clicking on a link in the tables below, you'll get tables with all polytopes for the corresponding group. You will also find files containing the involutions generating the polytopes that appear in this Atlas. For instance, if you want to reconstruct the polytope of Schlafli type {3,5} for the group Alt(5), click on the link below corresponding to Alt(5), then download the Magma file in the Alt(5) page. Once in Magma, load this file and take the entry number 1 in the sequence invols. It contains three involutions that generate Alt(5). Taken two by two, these involutions generate three subgroups of Alt(5) which are the maximal parabolic subgroups of a coset geometry which has the {5,3} diagram.

The groups are divided in the following families :

• Sporadic groups and their automorphism groups
• Alternating groups and their automorphism groups
• Unitary groups and their automorphism groups
• Suzuki groups and their automorphism groups

## Sporadic groups and their automorphism groups

G Aut(G) Order of G Number of involutions Number of Polytopes
M11 M11 7920 165 0
M12 M12:2 95040 891 37 = 23+14
M12:2 M12:2 190080 1683 266 = 223+43
J1 J1 175560 1463 150 = 148+2
M22 M22:2 443510 1155 0
M22:2 M22:2 887040 2871 195 = 133+62
J2 J2:2 604800 2835 154 = 137+17
J2:2 J2:2 1209600 4635 452 = 368+82+2
O'N O'N:2 460815505920 2857239 ??? = ??? + 31

## Alternating groups and their automorphism groups

G Aut(G) Order of G Number of involutions Number of Polytopes
Alt(5) = PSL(2,4) = PSL(2,5) Sym(5) 60 15 2
Sym(5) Sym(5) 120 25 5 = 4+1
Alt(6) = PSL(2,9) PΓL(2,9) 360 45 0
PGL(2,9) PΓL(2,9) 720 81 14
PΣL(2,9) PΓL(2,9) 720 75 7 = 2+4+1
M10 PΓL(2,9) 720 45 0
PΓL(2,9) PΓL(2,9) 1440 111 12
Alt(7) Sym(7) 2520 105 0
Sym(7) Sym(7) 5040 231 44 = 35+7+1+1
Alt(8) Sym(8) 20160 315 0
Sym(8) Sym(8) 40320 763 117 = 68+36+11+1+1
Alt(9) Sym(9) 181440 1323 47 = 41+6
Sym(9) Sym(9) 362880 2619 182 = 129+37+7+7+1+1

## Linear groups and their automorphism groups

G Aut(G) Order of G Number of involutions Number of Polytopes
Alt(5) = PSL(2,4) = PSL(2,5) Sym(5) 60 15 2
Sym(5) Sym(5) 120 25 5 = 4+1
PSL(3,2) = PSL(2,7) PΓL(2,7) 168 21 0
PGL(2,7) = PΓL(2,7) PΓL(2,7) 336 49 16
Alt(6) = PSL(2,9) PΓL(2,9) 360 45 0
PGL(2,9) PΓL(2,9) 720 81 14
PΣL(2,9) PΓL(2,9) 720 75 7 = 2+4+1
M10 PΓL(2,9) 720 45 0
PΓL(2,9) PΓL(2,9) 1440 111 12
PSL(2,8) = PGL(2,8) PΓL(2,8) 504 63 7
PΓL(2,8) = PΣL(2,8) PΓL(2,8) 1512 63 0
PSL(2,11) = PΣL(2,11) PΓL(2,11) 660 55 4 = 3+1
PGL(2,11) = PΓL(2,11) PΓL(2,11) 1320 121 42
PSL(2,13) = PΣL(2,13) PΓL(2,13) 1092 91 11
PGL(2,13) = PΓL(2,13) PΓL(2,13) 2184 169 59
PSL(2,17) = PΣL(2,17) PΓL(2,17) 2448 153 16
PGL(2,17) = PΓL(2,17) PΓL(2,17) 4896 289 110
PSL(2,19) = PΣL(2,19) PΓL(2,19) 3420 171 18 = 17+1
PGL(2,19) = PΓL(2,19) PΓL(2,19) 6840 361 140
PSL(2,16) = PGL(2,16) PΓL(2,16) 4080 255 27
PSL(2,16):2 PΓL(2,16) 8160 323 26 = 21+5
PΓL(2,16) = PΣL(2,16) PΓL(2,16) 16320 323 0
PSL(3,3) = PGL(3,3) = PΣL(3,3) = PΓL(3,3) PSL(3,3):2 5616 117 0
PSL(3,3):2 PSL(3,3):2 11232 351 68 = 67+1
PSL(2,23) = PΣL(2,23) PΓL(2,23) 6072 253 28
PGL(2,23) = PΓL(2,23) PΓL(2,23) 12144 529 212
PSL(2,25) PΓL(2,25) 7800 325 17
PGL(2,25) PΓL(2,25) 15600 625 127
PΣL(2,25) PΓL(2,25) 15600 455 51 = 34+17
PSL(2,25).2 PΓL(2,25) 7800 325 0
PΓL(2,25) PΓL(2,25) 31200 755 64
PSL(2,27) PΓL(2,27) 9828 351 14
PGL(2,27) PΓL(2,27) 19656 729 98
PΣL(2,27) PΓL(2,27) 29484 351 0
PΓL(2,27) PΓL(2,27) 58968 729 0
PSL(2,29) = PΣL(2,29) PΓL(2,29) 12180 435 50
PGL(2,29) = PΓL(2,29) PΓL(2,29) 24360 841 337
PSL(2,31) = PΣL(2,31) PΓL(2,31) 14880 465 51
PGL(2,31) = PΓL(2,31) PΓL(2,31) 29760 961 394
PSL(3,4) PSL(3,4).D12 20160 315 0
PSL(3,4).21 PSL(3,4).D12 40320 595 4
PSL(3,4).3 = PGL(3,4) PSL(3,4).D12 60480 315 0
PSL(3,4).3.23 PSL(3,4).D12 120960 1323 52 = 50+2
PSL(3,4).3.22 = PΓL(3,4) PSL(3,4).D12 120960 675 0
PSL(3,4).6 PSL(3,4).D12 120960 595 0
PSL(3,4).D12 PSL(3,4).D12 241920 1963 119 = 100+16+3
PSL(3,4).23 PSL(3,4).22 40320 651 53 = 44+9
PSL(3,4).22 = PΣL(3,4) PSL(3,4).22 40320 435 0
PSL(3,4).22 PSL(3,4).22 80640 1051 147 = 88+59
PSL(2,32) = PGL(2,32) PΓL(2,32) 32736 1023 93
PΓL(2,32) = PΣL(2,32) PΓL(2,32) 163680 1023 0
PSL(3,5) = PΣL(3,5) = PGL(3,5) = PΓL(3,5) PSL(3,5):2 372000 775 0
PSL(3,5):2 PSL(3,5):2 744000 3875 498 = 496+2
PSL(4,3) PGL(4,3):2 6065280 7371 18 = 9+9

## Unitary groups and their automorphism groups

G Aut(G) Order of G Number of involutions Number of Polytopes
PSU(3,3) = PGU(3,3) PΓL(3,3) 6048 63 0
PΓU(3,3) = PΣU(3,3) PΓU(3,3) 12096 315 31 = 25+6
PSU(4,2) = PGU(4,2) PΓU(4,2) 25920 315 0
PΓU(4,2) = PΣU(4,2) PΓU(4,2) 51840 891 147 = 87+50+10
PSU(3,4) = PGU(3,4) PΓU(3,4) 62400 195 0
PSU(3,4):2 PΓU(3,4) 124800 1235 80 = 78+2
PΣU(3,4) = PΓU(3,4) PΓU(3,4) 249600 1235 0
PSU(3,5) PΓU(3,5) 126000 525 0
PGU(3,5) PΓU(3,5) 378000 525 0
PΓU(3,5) PΓU(3,5) 756000 3675 247 = 237+10
PΣU(3,5) PΣU(3,5) 252000 1575 116 = 105+11

## Suzuki groups and their automorphism groups

G Aut(G) Order of G Number of involutions Number of Polytopes
Sz(8) Sz(8):3 29120 455 7
Sz(8):3 Sz(8):3 87360 455 0