# An Atlas of Chiral Polytopes for Small Almost Simple Groups

by Michael Hartley, Isabel Hubard and Dimitri Leemans

This Atlas contains all chiral 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 generators of the polytopes that appear in this Atlas.

The groups are divided in the following families :

• Sporadic groups and their automorphism groups
• Alternating groups and their automorphism groups
• Linear 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 Polytopes
M11 M11 7920 66 = 66
M12 M12:2 95040 184 = 118 + 64 + 2
M12:2 M12:2 190080 700 = 608 + 92
J1 J1 175560 1096 = 1056 + 40
M22 M22:2 443510 242 = 242
M22:2 M22:2 887040 1506 = 1442 + 64
J2 J2:2 604800 986 = 888 + 98
J2:2 J2:2 1209600 2218 = 1902 + 314 + 2

## Alternating groups and their automorphism groups

G Aut(G) Order of G Number of Polytopes
Alt(5) = PSL(2,4) = PSL(2,5) Sym(5) 60 0
Sym(5) = PGL(2,5) Sym(5) 120 6 = 0 + 6
Alt(6) = PSL(2,9) PΓL(2,9) 360 0
PGL(2,9) PΓL(2,9) 720 2 = 0 + 2
Sym(6) = PΣL(2,9) PΓL(2,9) 720 4 = 2 + 0 + 2
M10 PΓL(2,9) 720 0
PΓL(2,9) PΓL(2,9) 1440 28 = 8 + 20
Alt(7) Sym(7) 2520 0
Sym(7) Sym(7) 5040 102 = 50 + 52
Alt(8) Sym(8) 20160 14 = 14
Sym(8) Sym(8) 40320 238 = 182 + 48 + 8
Alt(9) Sym(9) 181440 348 = 270 + 78
Sym(9) Sym(9) 362880 968 = 836 + 132

## Linear groups and their automorphism groups

G Aut(G) Order of G Number of Polytopes
Alt(5) = PSL(2,4) = PSL(2,5) Sym(5) 60 0
Sym(5) = PGL(2,5) Sym(5) 120 6 = 0 + 6
PSL(3,2) = PSL(2,7) PΓL(2,7) 168 0
PGL(2,7) = PΓL(2,7) PΓL(2,7) 336 10 = 0 + 10
Alt(6) = PSL(2,9) PΓL(2,9) 360 0
PGL(2,9) PΓL(2,9) 720 2 = 0 + 2
Sym(6) = PΣL(2,9) PΓL(2,9) 720 4 = 2 + 0 + 2
M10 PΓL(2,9) 720 0
PΓL(2,9) PΓL(2,9) 1440 28 = 8 + 20
PSL(2,8) = PGL(2,8) PΓL(2,8) 504 2 = 0 + 2
PΓL(2,8) = PΣL(2,8) PΓL(2,8) 1512 28 = 28
PSL(2,11) = PΣL(2,11) PΓL(2,11) 660 0
PGL(2,11) = PΓL(2,11) PΓL(2,11) 1320 24 = 0 + 24
PSL(2,13) = PΣL(2,13) PΓL(2,13) 1092 6 = 0 + 6
PGL(2,13) = PΓL(2,13) PΓL(2,13) 2184 14 = 0 + 14
PSL(2,17) = PΣL(2,17) PΓL(2,17) 2448 10 = 0 + 10
PGL(2,17) = PΓL(2,17) PΓL(2,17) 4896 8 = 0 + 8
PSL(2,19) = PΣL(2,19) PΓL(2,19) 3420 4 = 0 + 4
PGL(2,19) = PΓL(2,19) PΓL(2,19) 6840 28 = 0 + 28
PSL(2,16) = PGL(2,16) PΓL(2,16) 4080 2 = 0 + 2
PSL(2,16):2 PΓL(2,16) 8160 48 = 32 + 16
PΓL(2,16) = PΣL(2,16) PΓL(2,16) 16320 122 = 122
PSL(3,3) = PGL(3,3) = PΣL(3,3) = PΓL(3,3) PSL(3,3):2 5616 0
PSL(3,3):2 PSL(3,3):2 11232 168 = 136 + 32
PSL(2,23) = PΣL(2,23) PΓL(2,23) 6072 0
PGL(2,23) = PΓL(2,23) PΓL(2,23) 12144 10 = 0 + 10
PSL(2,25) PΓL(2,25) 7800 2 = 0 + 2
PGL(2,25) PΓL(2,25) 15600 6 = 0 + 6
PΣL(2,25) PΓL(2,25) 15600 62 = 38 + 24
PSL(2,25).2 PΓL(2,25) 15600 30 = 30
PΓL(2,25) PΓL(2,25) 31200 152 = 108 + 44
PSL(2,27) PΓL(2,27) 9828 0
PGL(2,27) PΓL(2,27) 19656 4 = 0 + 4
PΣL(2,27) PΓL(2,27) 29484 108 = 108
PΓL(2,27) PΓL(2,27) 58968 324 = 324
PSL(2,29) = PΣL(2,29) PΓL(2,29) 12180 10 = 0 + 10
PGL(2,29) = PΓL(2,29) PΓL(2,29) 24360 26 = 0 + 26
PSL(2,31) = PΣL(2,31) PΓL(2,31) 14880 6 = 0 + 6
PGL(2,31) = PΓL(2,31) PΓL(2,31) 29760 46 = 0 + 46
PSL(3,4) PSL(3,4).D12 20160 0
PSL(3,4).21 PSL(3,4).D12 40320 28 = 24 + 4
PSL(3,4).3 = PGL(3,4) PSL(3,4).D12 60480 0
PSL(3,4).3.23 PSL(3,4).D12 120960 262 = 224 + 38
PSL(3,4).3.22 = PΓL(3,4) PSL(3,4).D12 120960 24 = 24
PSL(3,4).6 PSL(3,4).D12 120960 60 = 60
PSL(3,4).D12 PSL(3,4).D12 241920 392 = 296 + 88 + 8
PSL(3,4).23 PSL(3,4).22 40320 138 = 102 + 36
PSL(3,4).22 = PΣL(3,4) PSL(3,4).22 40320 30 = 26 + 4
PSL(3,4).22 PSL(3,4).22 80640 216 = 164 + 52
PSL(2,32) = PGL(2,32) PΓL(2,32) 32736 6 = 0 + 6
PΓL(2,32) = PΣL(2,32) PΓL(2,32) 163680 744
PSL(3,5) = PΣL(3,5) = PGL(3,5) = PΓL(3,5) PSL(3,5):2 372000 2 = 0 + 2
PSL(3,5):2 PSL(3,5):2 744000 2668 = 2494 + 174

## Unitary groups and their automorphism groups

G Aut(G) Order of G Number of Polytopes
PSU(3,3) = PGU(3,3) PΓL(3,3) 6048 0
PΓU(3,3) = PΣU(3,3) PΓU(3,3) 12096 166 = 146 + 20
PSU(4,2) = PGU(4,2) PΓU(4,2) 25920 26 = 26
PΓU(4,2) = PΣU(4,2) PΓU(4,2) 51840 370 = 270 + 100
PSU(3,4) = PGU(3,4) PΓU(3,4) 62400 0
PSU(3,4):2 PΓU(3,4) 124800 418 = 376 + 42
PΣU(3,4) = PΓU(3,4) PΓU(3,4) 249600 526 = 526
PSU(3,5) PΓU(3,5) 126000 0
PGU(3,5) PΓU(3,5) 378000 0
PΓU(3,5) PΓU(3,5) 756000 1754 = 1580 + 174
PΣU(3,5) PΣU(3,5) 252000 962 = 834 + 120 + 8

## Suzuki groups and their automorphism groups

G Aut(G) Order of G Number of Polytopes
Sz(8) Sz(8):3 29120 128 = 128
Sz(8):3 Sz(8):3 87360 284 = 284