Finite Group DatabasebetaFinite Group Databasebeta| order | 10752 |
| isoclinism | isoclinic class of #72209665 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini, Fitting, cosocle | C26 | GL(3,𝔽2) |
| socle | C23 | #13051525 |
| LCS | G | |
| UCS | C1 | |
372 conjugacy classes of subgroups; only the 4 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 4 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(1,2,5,3)(4,18,20,19)(6,17,9,13)(7,11,26,14)(8,16,10,12)(15,27,21,28)(22,31,23,35)(24,32,25,36)(29,41,53,42)(30,46,56,47)(33,51,45,52)(34,54,50,55)(37,57,43,58)(38,59,48,60)(39,61,44,62)(40,63,49,64);(1,4,2)(3,6,15)(5,25,16)(7,20,17)(8,21,13)(9,23,12)(10,22,14)(11,26,24)(18,29,30)(19,33,34)(27,37,38)(28,39,40)(31,45,49)(32,44,50)(35,53,48)(36,43,56)(42,51,62)(47,60,64)(52,58,61)(54,63,59);(1,5)(2,11)(3,14)(4,20)(6,9)(7,26)(8,10)(12,13)(15,21)(16,17)(18,31)(19,35)(22,23)(24,25)(27,32)(28,36)(29,33)(30,48)(34,49)(37,39)(38,56)(40,50)(41,58)(42,57)(43,44)(45,53)(46,63)(47,64)(51,62)(52,61)(54,59)(55,60);(1,6)(2,12)(3,16)(4,21)(5,9)(7,8)(10,26)(11,13)(14,17)(15,20)(18,32)(19,36)(22,25)(23,24)(27,31)(28,35)(29,43)(30,38)(33,44)(34,40)(37,53)(39,45)(41,51)(42,52)(46,64)(47,63)(48,56)(49,50)(54,60)(55,59)(57,61)(58,62);(1,7)(2,3)(4,22)(5,26)(6,8)(9,10)(11,14)(12,16)(13,17)(15,24)(18,19)(20,23)(21,25)(27,28)(29,43)(30,40)(31,35)(32,36)(33,44)(34,38)(37,53)(39,45)(41,51)(42,52)(46,60)(47,59)(48,50)(49,56)(54,64)(55,63)(57,61)(58,62);(1,8)(2,11)(3,14)(4,23)(5,10)(6,7)(9,26)(12,13)(15,25)(16,17)(18,27)(19,28)(20,22)(21,24)(29,44)(30,49)(31,32)(33,43)(34,48)(35,36)(37,45)(38,50)(39,53)(40,56)(41,58)(42,57)(46,54)(47,55)(51,62)(52,61)(59,63)(60,64);(1,9)(2,13)(3,17)(4,24)(5,6)(7,10)(8,26)(11,12)(14,16)(15,22)(18,32)(19,36)(20,25)(21,23)(27,31)(28,35)(29,45)(30,38)(33,53)(34,40)(37,44)(39,43)(41,52)(42,51)(46,59)(47,60)(48,56)(49,50)(54,63)(55,64)(57,62)(58,61);(1,10)(2,14)(3,11)(4,22)(5,8)(6,26)(7,9)(12,17)(13,16)(15,24)(18,28)(19,27)(20,23)(21,25)(29,39)(30,40)(31,36)(32,35)(33,37)(34,38)(41,57)(42,58)(43,45)(44,53)(46,55)(47,54)(48,50)(49,56)(51,61)(52,62)(59,64)(60,63)24 conjugacy classes — view the full character table.
| maximal quotient of | #72209920 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,2,5,3)(4,18,20,19)(6,17,9,13)(7,11,26,14)(8,16,10,12)(15,27,21,28)(22,31,23,35)(24,32,25,36)(29,41,53,42)(30,46,56,47)(33,51,45,52)(34,54,50,55)(37,57,43,58)(38,59,48,60)(39,61,44,62)(40,63,49,64), (1,4,2)(3,6,15)(5,25,16)(7,20,17)(8,21,13)(9,23,12)(10,22,14)(11,26,24)(18,29,30)(19,33,34)(27,37,38)(28,39,40)(31,45,49)(32,44,50)(35,53,48)(36,43,56)(42,51,62)(47,60,64)(52,58,61)(54,63,59), (1,5)(2,11)(3,14)(4,20)(6,9)(7,26)(8,10)(12,13)(15,21)(16,17)(18,31)(19,35)(22,23)(24,25)(27,32)(28,36)(29,33)(30,48)(34,49)(37,39)(38,56)(40,50)(41,58)(42,57)(43,44)(45,53)(46,63)(47,64)(51,62)(52,61)(54,59)(55,60), (1,6)(2,12)(3,16)(4,21)(5,9)(7,8)(10,26)(11,13)(14,17)(15,20)(18,32)(19,36)(22,25)(23,24)(27,31)(28,35)(29,43)(30,38)(33,44)(34,40)(37,53)(39,45)(41,51)(42,52)(46,64)(47,63)(48,56)(49,50)(54,60)(55,59)(57,61)(58,62), (1,7)(2,3)(4,22)(5,26)(6,8)(9,10)(11,14)(12,16)(13,17)(15,24)(18,19)(20,23)(21,25)(27,28)(29,43)(30,40)(31,35)(32,36)(33,44)(34,38)(37,53)(39,45)(41,51)(42,52)(46,60)(47,59)(48,50)(49,56)(54,64)(55,63)(57,61)(58,62), (1,8)(2,11)(3,14)(4,23)(5,10)(6,7)(9,26)(12,13)(15,25)(16,17)(18,27)(19,28)(20,22)(21,24)(29,44)(30,49)(31,32)(33,43)(34,48)(35,36)(37,45)(38,50)(39,53)(40,56)(41,58)(42,57)(46,54)(47,55)(51,62)(52,61)(59,63)(60,64), (1,9)(2,13)(3,17)(4,24)(5,6)(7,10)(8,26)(11,12)(14,16)(15,22)(18,32)(19,36)(20,25)(21,23)(27,31)(28,35)(29,45)(30,38)(33,53)(34,40)(37,44)(39,43)(41,52)(42,51)(46,59)(47,60)(48,56)(49,50)(54,63)(55,64)(57,62)(58,61), (1,10)(2,14)(3,11)(4,22)(5,8)(6,26)(7,9)(12,17)(13,16)(15,24)(18,28)(19,27)(20,23)(21,25)(29,39)(30,40)(31,36)(32,35)(33,37)(34,38)(41,57)(42,58)(43,45)(44,53)(46,55)(47,54)(48,50)(49,56)(51,61)(52,62)(59,64)(60,63));