An Atlas of Polytopes for Small Almost Simple Groups

by Laurence Vauthier and Dimitri Leemans

with sporadic group O'Nan added by Thomas Connor, Leemans and Mark Mixer

with sporadic groups from M23 excluding O'Nan added by Leemans and Jessica Mulpas

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
    M23 M23 10200960 3795 0
    HS HS:2 44352000 21175 311 = 252+57+2
    J3 J3:2 50232960 26163 305 = 303+2
    M24 M24 244823040 43263 647 = 490+155+2
    McL McL:2 898128000 22275 0
    He He:2 4,030,387,200 212415 1264 = 1188+76
    Ru Ru 145,926,144,000 1846575 21821 = 21594+227
    Suz Suz:2 448,345,497,600 2915055 7389 = 7119+257+13
    O'N O'N:2 460,815,505,920 2857239 6552 = 6536 + 16
    Co3 Co3 495,766,656,000 2778975 11481 = 10586+873+22

    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
    Sym(6) = 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
    Sym(10) Sym(10) 3628800 9495 690=413+203+52+13+7+1+1
    Sym(11) Sym(11) 39916800 35695 1496=1221+189+43+25+9+7+1+1
    Sym(12) Sym(12) 479001600 140151 4602=3346+940+183+75+40+9+7+1+1
    Sym(13) Sym(13) 6227020800 568503 8414=7163+863+171+123+41+35+9+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