Finite Group DatabasebetaFinite Group Databasebeta| order | 10752 |
| isoclinism | isoclinic class of #72209652 |
| 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 | C43 | GL(3,𝔽2) |
| socle | C23 | #13051525 |
| LCS | G | |
| UCS | C1 | |
213 conjugacy classes of subgroups; only the 4 normal subgroups are shown.
This group has 213 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)(3,10)(4,8)(5,9)(6,11)(7,15)(12,22)(13,21)(14,23)(16,27)(17,26)(18,28)(19,29)(20,33)(24,37)(25,41)(30,47)(31,46)(32,48)(34,52)(35,53)(36,51)(38,57)(39,56)(40,58)(42,59)(43,60)(44,61)(49,65)(50,69)(54,73)(55,77)(62,63)(66,85)(67,83)(68,84)(70,90)(71,88)(72,89)(74,93)(75,94)(76,95)(78,82)(79,97)(80,98)(81,96)(86,101)(87,91)(92,110)(99,100)(102,103)(104,108)(105,107)(106,109);(2,6,7)(8,12,17)(9,13,16)(10,14,18)(11,19,20)(15,24,25)(21,31,34)(22,30,35)(23,32,36)(26,38,42)(27,39,43)(28,40,44)(29,45,41)(33,49,50)(37,54,55)(46,62,59)(47,63,60)(48,64,61)(51,66,70)(52,67,71)(53,68,72)(56,74,78)(57,75,79)(58,76,80)(65,81,82)(69,86,87)(73,91,92)(77,84,96)(83,99,98)(85,100,97)(88,101,105)(89,102,104)(90,103,106)(93,107,111)(94,108,112)(95,109,110);(1,3)(2,8,10,9)(4,5)(7,16,18,17)(19,30,32,31)(20,34,36,35)(24,38,40,39)(25,42,44,43)(33,51)(37,56,58,57)(41,59,61,60)(45,62,64,63)(49,66)(50,70)(52,53)(54,74,76,75)(65,83,85,84)(67,68)(69,88,90,89)(71,72)(73,93,95,94)(77,97,98,82)(87,104,106,105)(92,111,110,112);(1,3)(2,8,10,9)(4,5)(6,12,14,13)(7,18)(11,21,23,22)(15,26,28,27)(16,17)(19,32)(20,34,36,35)(24,40)(25,43,44,42)(29,46,48,47)(30,31)(33,51)(37,58)(38,39)(41,60,61,59)(45,64)(49,66)(50,70)(52,53)(54,75,76,74)(55,78,80,79)(56,57)(62,63)(65,83,85,84)(67,68)(69,88,90,89)(71,72)(73,94,95,93)(77,98)(81,99,100,96)(82,97)(86,101,103,102)(87,106)(91,107,109,108)(92,110)(104,105)(111,112);(1,4,3,5)(6,14)(11,23)(12,13)(15,26,28,27)(21,22)(24,39,40,38)(25,43,44,42)(29,46,48,47)(33,52,51,53)(37,57,58,56)(41,60,61,59)(45,63,64,62)(49,67,66,68)(50,71,70,72)(54,75,76,74)(55,78,80,79)(73,94,95,93)(81,100)(86,103)(91,107,109,108)(92,112,110,111)(96,99)(101,102);(2,10)(7,18)(8,9)(16,17)(19,32)(20,36)(24,40)(25,44)(30,31)(34,35)(37,58)(38,39)(41,61)(42,43)(45,64)(54,76)(56,57)(59,60)(62,63)(65,85)(69,90)(73,95)(74,75)(77,98)(82,97)(83,84)(87,106)(88,89)(92,110)(93,94)(104,105)(111,112);(2,10)(6,14)(8,9)(11,23)(12,13)(15,28)(20,36)(21,22)(25,44)(26,27)(29,48)(34,35)(41,61)(42,43)(46,47)(54,76)(55,80)(59,60)(65,85)(69,90)(73,95)(74,75)(78,79)(81,100)(83,84)(86,103)(88,89)(91,109)(93,94)(96,99)(101,102)(107,108);(1,3)(4,5)(15,28)(24,40)(25,44)(26,27)(29,48)(33,51)(37,58)(38,39)(41,61)(42,43)(45,64)(46,47)(49,66)(50,70)(52,53)(54,76)(55,80)(56,57)(59,60)(62,63)(67,68)(71,72)(73,95)(74,75)(78,79)(91,109)(92,110)(93,94)(107,108)(111,112)18 conjugacy classes — view the full character table.
| maximal quotient of | #72209828 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,2)(3,10)(4,8)(5,9)(6,11)(7,15)(12,22)(13,21)(14,23)(16,27)(17,26)(18,28)(19,29)(20,33)(24,37)(25,41)(30,47)(31,46)(32,48)(34,52)(35,53)(36,51)(38,57)(39,56)(40,58)(42,59)(43,60)(44,61)(49,65)(50,69)(54,73)(55,77)(62,63)(66,85)(67,83)(68,84)(70,90)(71,88)(72,89)(74,93)(75,94)(76,95)(78,82)(79,97)(80,98)(81,96)(86,101)(87,91)(92,110)(99,100)(102,103)(104,108)(105,107)(106,109), (2,6,7)(8,12,17)(9,13,16)(10,14,18)(11,19,20)(15,24,25)(21,31,34)(22,30,35)(23,32,36)(26,38,42)(27,39,43)(28,40,44)(29,45,41)(33,49,50)(37,54,55)(46,62,59)(47,63,60)(48,64,61)(51,66,70)(52,67,71)(53,68,72)(56,74,78)(57,75,79)(58,76,80)(65,81,82)(69,86,87)(73,91,92)(77,84,96)(83,99,98)(85,100,97)(88,101,105)(89,102,104)(90,103,106)(93,107,111)(94,108,112)(95,109,110), (1,3)(2,8,10,9)(4,5)(7,16,18,17)(19,30,32,31)(20,34,36,35)(24,38,40,39)(25,42,44,43)(33,51)(37,56,58,57)(41,59,61,60)(45,62,64,63)(49,66)(50,70)(52,53)(54,74,76,75)(65,83,85,84)(67,68)(69,88,90,89)(71,72)(73,93,95,94)(77,97,98,82)(87,104,106,105)(92,111,110,112), (1,3)(2,8,10,9)(4,5)(6,12,14,13)(7,18)(11,21,23,22)(15,26,28,27)(16,17)(19,32)(20,34,36,35)(24,40)(25,43,44,42)(29,46,48,47)(30,31)(33,51)(37,58)(38,39)(41,60,61,59)(45,64)(49,66)(50,70)(52,53)(54,75,76,74)(55,78,80,79)(56,57)(62,63)(65,83,85,84)(67,68)(69,88,90,89)(71,72)(73,94,95,93)(77,98)(81,99,100,96)(82,97)(86,101,103,102)(87,106)(91,107,109,108)(92,110)(104,105)(111,112), (1,4,3,5)(6,14)(11,23)(12,13)(15,26,28,27)(21,22)(24,39,40,38)(25,43,44,42)(29,46,48,47)(33,52,51,53)(37,57,58,56)(41,60,61,59)(45,63,64,62)(49,67,66,68)(50,71,70,72)(54,75,76,74)(55,78,80,79)(73,94,95,93)(81,100)(86,103)(91,107,109,108)(92,112,110,111)(96,99)(101,102), (2,10)(7,18)(8,9)(16,17)(19,32)(20,36)(24,40)(25,44)(30,31)(34,35)(37,58)(38,39)(41,61)(42,43)(45,64)(54,76)(56,57)(59,60)(62,63)(65,85)(69,90)(73,95)(74,75)(77,98)(82,97)(83,84)(87,106)(88,89)(92,110)(93,94)(104,105)(111,112), (2,10)(6,14)(8,9)(11,23)(12,13)(15,28)(20,36)(21,22)(25,44)(26,27)(29,48)(34,35)(41,61)(42,43)(46,47)(54,76)(55,80)(59,60)(65,85)(69,90)(73,95)(74,75)(78,79)(81,100)(83,84)(86,103)(88,89)(91,109)(93,94)(96,99)(101,102)(107,108), (1,3)(4,5)(15,28)(24,40)(25,44)(26,27)(29,48)(33,51)(37,58)(38,39)(41,61)(42,43)(45,64)(46,47)(49,66)(50,70)(52,53)(54,76)(55,80)(56,57)(59,60)(62,63)(67,68)(71,72)(73,95)(74,75)(78,79)(91,109)(92,110)(93,94)(107,108)(111,112));