Finite Group DatabasebetaFinite Group Databasebeta| order | 15360 |
| isoclinism | isoclinic class of #72209693 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #62694 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C42 | #11757162 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | #62694 | A5 |
| socle | C22 | #72209602 |
| LCS | G | |
| UCS | C1 ◁ C42 | |
1392 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 1392 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(2,8)(3,5)(4,22)(9,35)(10,38)(11,28)(13,18)(14,27)(15,26)(16,54)(17,56)(19,30)(20,31)(21,32)(23,40)(24,61)(25,29)(36,37)(39,57)(41,51)(42,53)(43,59)(44,63)(45,62)(46,58)(47,48)(52,60)(55,64)(66,68)(67,69)(72,93)(73,95)(74,85)(75,86)(76,81)(77,87)(78,88)(80,108)(82,89)(83,90)(84,91)(94,121)(96,117)(98,114)(99,116)(100,111)(101,107)(102,112)(103,120)(104,109)(105,110)(106,115)(118,119)(122,128)(123,126)(125,127);(2,9,10)(3,16,17)(4,23,24)(5,8,28)(11,12,42)(13,51,43)(14,19,41)(15,30,36)(18,45,49)(20,39,48)(21,57,59)(22,35,44)(25,46,50)(26,58,32)(27,62,31)(29,37,47)(34,60,40)(38,64,53)(52,63,56)(54,55,61)(66,72,73)(67,79,80)(68,85,69)(74,101,81)(75,104,98)(76,94,108)(77,89,99)(78,82,100)(83,97,106)(84,113,112)(86,117,103)(87,114,90)(88,120,91)(93,121,110)(95,122,123)(96,116,102)(105,127,118)(107,126,125)(109,111,115)(119,124,128);(1,2)(3,11)(4,12)(5,29)(6,14)(7,15)(8,34)(9,36)(10,19)(13,24)(16,37)(17,58)(18,43)(20,45)(21,46)(22,33)(23,59)(25,47)(26,49)(27,50)(28,48)(30,44)(31,53)(32,52)(35,41)(38,39)(40,61)(42,64)(51,54)(55,62)(56,60)(57,63)(65,66)(67,74)(68,86)(69,76)(70,77)(71,78)(72,73)(75,105)(79,111)(80,114)(81,115)(82,100)(83,98)(84,103)(85,112)(87,101)(88,107)(89,99)(90,109)(91,96)(92,110)(93,121)(94,113)(95,97)(102,126)(104,122)(106,127)(108,117)(116,124)(118,123)(119,125)(120,128);(1,3)(2,11)(4,25)(5,30)(6,20)(7,21)(8,18)(9,31)(10,13)(12,47)(14,45)(15,46)(16,55)(17,54)(19,24)(22,40)(23,38)(26,42)(27,60)(28,63)(29,44)(32,35)(33,61)(34,43)(36,53)(37,62)(39,59)(41,52)(48,57)(49,64)(50,56)(51,58)(65,67)(66,74)(68,81)(69,89)(70,83)(71,84)(72,91)(73,96)(75,102)(76,99)(77,98)(78,103)(79,112)(80,97)(82,118)(85,111)(86,115)(87,104)(88,117)(90,93)(92,119)(94,120)(95,114)(100,123)(101,122)(105,126)(106,124)(107,108)(109,121)(110,125)(113,128)(116,127);(1,4)(2,12)(3,18)(5,23)(6,26)(7,27)(8,33)(9,37)(10,39)(11,43)(13,48)(14,49)(15,50)(16,36)(17,53)(19,38)(20,60)(21,42)(22,34)(24,28)(25,61)(29,59)(30,63)(31,58)(32,62)(35,51)(40,47)(41,54)(44,57)(45,56)(46,64)(52,55)(65,68)(66,75)(67,81)(69,85)(70,87)(71,88)(72,94)(73,97)(74,102)(76,106)(77,107)(78,101)(79,100)(80,96)(82,116)(83,104)(84,117)(86,110)(89,111)(90,114)(91,120)(92,105)(93,95)(98,108)(99,124)(103,122)(109,128)(112,123)(113,121)(115,125)(118,127)(119,126);(1,5)(2,13)(3,19)(4,23)(6,31)(7,32)(8,28)(9,21)(10,40)(11,44)(12,48)(14,52)(15,53)(16,56)(17,50)(18,38)(20,35)(22,29)(24,33)(25,63)(26,58)(27,62)(30,61)(34,59)(36,45)(37,42)(39,47)(41,46)(43,57)(49,55)(51,60)(54,64)(65,69)(66,76)(67,82)(68,85)(70,90)(71,91)(72,83)(73,98)(74,100)(75,106)(77,109)(78,96)(79,102)(80,101)(81,116)(84,93)(86,112)(87,114)(88,120)(89,119)(92,118)(94,104)(95,117)(97,108)(99,125)(103,121)(105,127)(107,128)(110,123)(111,126)(113,122)(115,124);(65,70,92,71)(66,77,110,78)(67,83,119,84)(68,87,105,88)(69,90,118,91)(72,89,93,82)(73,99,121,100)(74,98,125,103)(75,107,86,101)(76,109,123,96)(79,97,124,113)(80,106,128,112)(81,104,126,117)(85,114,127,120)(94,111,95,116)(102,108,115,122);(1,6,33,7)(2,14,22,15)(3,20,61,21)(4,26,8,27)(5,31,24,32)(9,19,35,30)(10,41,44,36)(11,45,40,46)(12,49,34,50)(13,52,29,53)(16,39,54,57)(17,48,55,59)(18,60,25,42)(23,58,28,62)(37,38,51,63)(43,56,47,64)84 conjugacy classes — view the full character table.
| maximal quotient of | #72209980 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((2,8)(3,5)(4,22)(9,35)(10,38)(11,28)(13,18)(14,27)(15,26)(16,54)(17,56)(19,30)(20,31)(21,32)(23,40)(24,61)(25,29)(36,37)(39,57)(41,51)(42,53)(43,59)(44,63)(45,62)(46,58)(47,48)(52,60)(55,64)(66,68)(67,69)(72,93)(73,95)(74,85)(75,86)(76,81)(77,87)(78,88)(80,108)(82,89)(83,90)(84,91)(94,121)(96,117)(98,114)(99,116)(100,111)(101,107)(102,112)(103,120)(104,109)(105,110)(106,115)(118,119)(122,128)(123,126)(125,127), (2,9,10)(3,16,17)(4,23,24)(5,8,28)(11,12,42)(13,51,43)(14,19,41)(15,30,36)(18,45,49)(20,39,48)(21,57,59)(22,35,44)(25,46,50)(26,58,32)(27,62,31)(29,37,47)(34,60,40)(38,64,53)(52,63,56)(54,55,61)(66,72,73)(67,79,80)(68,85,69)(74,101,81)(75,104,98)(76,94,108)(77,89,99)(78,82,100)(83,97,106)(84,113,112)(86,117,103)(87,114,90)(88,120,91)(93,121,110)(95,122,123)(96,116,102)(105,127,118)(107,126,125)(109,111,115)(119,124,128), (1,2)(3,11)(4,12)(5,29)(6,14)(7,15)(8,34)(9,36)(10,19)(13,24)(16,37)(17,58)(18,43)(20,45)(21,46)(22,33)(23,59)(25,47)(26,49)(27,50)(28,48)(30,44)(31,53)(32,52)(35,41)(38,39)(40,61)(42,64)(51,54)(55,62)(56,60)(57,63)(65,66)(67,74)(68,86)(69,76)(70,77)(71,78)(72,73)(75,105)(79,111)(80,114)(81,115)(82,100)(83,98)(84,103)(85,112)(87,101)(88,107)(89,99)(90,109)(91,96)(92,110)(93,121)(94,113)(95,97)(102,126)(104,122)(106,127)(108,117)(116,124)(118,123)(119,125)(120,128), (1,3)(2,11)(4,25)(5,30)(6,20)(7,21)(8,18)(9,31)(10,13)(12,47)(14,45)(15,46)(16,55)(17,54)(19,24)(22,40)(23,38)(26,42)(27,60)(28,63)(29,44)(32,35)(33,61)(34,43)(36,53)(37,62)(39,59)(41,52)(48,57)(49,64)(50,56)(51,58)(65,67)(66,74)(68,81)(69,89)(70,83)(71,84)(72,91)(73,96)(75,102)(76,99)(77,98)(78,103)(79,112)(80,97)(82,118)(85,111)(86,115)(87,104)(88,117)(90,93)(92,119)(94,120)(95,114)(100,123)(101,122)(105,126)(106,124)(107,108)(109,121)(110,125)(113,128)(116,127), (1,4)(2,12)(3,18)(5,23)(6,26)(7,27)(8,33)(9,37)(10,39)(11,43)(13,48)(14,49)(15,50)(16,36)(17,53)(19,38)(20,60)(21,42)(22,34)(24,28)(25,61)(29,59)(30,63)(31,58)(32,62)(35,51)(40,47)(41,54)(44,57)(45,56)(46,64)(52,55)(65,68)(66,75)(67,81)(69,85)(70,87)(71,88)(72,94)(73,97)(74,102)(76,106)(77,107)(78,101)(79,100)(80,96)(82,116)(83,104)(84,117)(86,110)(89,111)(90,114)(91,120)(92,105)(93,95)(98,108)(99,124)(103,122)(109,128)(112,123)(113,121)(115,125)(118,127)(119,126), (1,5)(2,13)(3,19)(4,23)(6,31)(7,32)(8,28)(9,21)(10,40)(11,44)(12,48)(14,52)(15,53)(16,56)(17,50)(18,38)(20,35)(22,29)(24,33)(25,63)(26,58)(27,62)(30,61)(34,59)(36,45)(37,42)(39,47)(41,46)(43,57)(49,55)(51,60)(54,64)(65,69)(66,76)(67,82)(68,85)(70,90)(71,91)(72,83)(73,98)(74,100)(75,106)(77,109)(78,96)(79,102)(80,101)(81,116)(84,93)(86,112)(87,114)(88,120)(89,119)(92,118)(94,104)(95,117)(97,108)(99,125)(103,121)(105,127)(107,128)(110,123)(111,126)(113,122)(115,124), (65,70,92,71)(66,77,110,78)(67,83,119,84)(68,87,105,88)(69,90,118,91)(72,89,93,82)(73,99,121,100)(74,98,125,103)(75,107,86,101)(76,109,123,96)(79,97,124,113)(80,106,128,112)(81,104,126,117)(85,114,127,120)(94,111,95,116)(102,108,115,122), (1,6,33,7)(2,14,22,15)(3,20,61,21)(4,26,8,27)(5,31,24,32)(9,19,35,30)(10,41,44,36)(11,45,40,46)(12,49,34,50)(13,52,29,53)(16,39,54,57)(17,48,55,59)(18,60,25,42)(23,58,28,62)(37,38,51,63)(43,56,47,64));