Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| Fitting | C24×#585 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #72209703 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | C24×#585 | A5 |
| socle | C2 | #72219284 |
| LCS | G | |
| UCS | C1 ◁ C4 | |
At least 87762 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has at least 87762 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 160 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 160 points:
(1,72,55,78,108,13,136,104,99,35)(2,23,93,120,28,20,100,134,146,16)(3,61,96,17,97,30,42,22,29,137)(4,80,118,148,60,34,102,105,98,41)(5,40,81,50,54,38,51,111,139,8)(6,94,59,9,125,10,48,39,26,140)(7,132,135,142,63,11,67,130,128,45)(12,90,110,64,149,47,95,66,46,153)(14,56,49,27,151,75,25,21,58,157)(15,114,129,116,92,18,89,133,126,32)(19,82,79,70,138,31,112,87,33,155)(24,123,143,107,145,103,124,53,144,101)(36,83,109,43,160,84,113,77,62,159)(37,119,88,71,154,44,69,65,85,156)(52,115,141,86,57,127,91,158,76,106)(68,74,121,131,152,73,122,117,147,150);(1,4,13,34)(2,7,20,11)(3,10,30,6)(5,15,38,18)(8,24,54,103)(9,27,26,58)(12,19,47,31)(14,36,75,84)(16,41,28,60)(17,43,29,62)(21,22,49,96)(23,51,100,40)(25,57,56,106)(32,68,92,73)(33,71,70,85)(35,63,108,45)(37,76,44,86)(39,77,59,109)(42,82,61,112)(46,91,64,115)(48,95,94,90)(50,98,139,148)(52,79,127,87)(53,102,143,80)(55,105,104,118)(65,66,88,110)(67,89,132,114)(69,83,119,113)(72,121,136,117)(74,123,122,124)(78,126,99,116)(81,129,111,133)(93,135,134,130)(97,138,137,155)(101,120,145,146)(107,131,144,147)(125,153,140,149)(128,152,142,150)(141,157,158,151)(154,159,156,160);(1,2,5)(3,8,21)(4,11,15)(6,16,39)(7,18,34)(9,25,23)(10,28,59)(12,32,65)(13,20,38)(14,35,22)(17,42,40)(19,45,87)(24,52,76)(26,56,100)(27,48,72)(29,61,51)(30,54,49)(31,63,79)(33,69,67)(36,73,109)(37,41,66)(43,83,117)(44,60,110)(46,90,89)(47,92,88)(50,97,55)(53,91,57)(58,94,136)(62,113,121)(64,95,114)(68,77,84)(70,119,132)(71,82,80)(75,108,96)(78,125,93)(81,120,151)(85,112,102)(86,103,127)(98,138,135)(99,140,134)(101,141,144)(104,139,137)(105,126,153)(106,143,115)(107,145,158)(111,146,157)(116,149,118)(128,154,129)(130,148,155)(131,152,160)(133,142,156)(147,150,159)62 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,72,55,78,108,13,136,104,99,35)(2,23,93,120,28,20,100,134,146,16)(3,61,96,17,97,30,42,22,29,137)(4,80,118,148,60,34,102,105,98,41)(5,40,81,50,54,38,51,111,139,8)(6,94,59,9,125,10,48,39,26,140)(7,132,135,142,63,11,67,130,128,45)(12,90,110,64,149,47,95,66,46,153)(14,56,49,27,151,75,25,21,58,157)(15,114,129,116,92,18,89,133,126,32)(19,82,79,70,138,31,112,87,33,155)(24,123,143,107,145,103,124,53,144,101)(36,83,109,43,160,84,113,77,62,159)(37,119,88,71,154,44,69,65,85,156)(52,115,141,86,57,127,91,158,76,106)(68,74,121,131,152,73,122,117,147,150), (1,4,13,34)(2,7,20,11)(3,10,30,6)(5,15,38,18)(8,24,54,103)(9,27,26,58)(12,19,47,31)(14,36,75,84)(16,41,28,60)(17,43,29,62)(21,22,49,96)(23,51,100,40)(25,57,56,106)(32,68,92,73)(33,71,70,85)(35,63,108,45)(37,76,44,86)(39,77,59,109)(42,82,61,112)(46,91,64,115)(48,95,94,90)(50,98,139,148)(52,79,127,87)(53,102,143,80)(55,105,104,118)(65,66,88,110)(67,89,132,114)(69,83,119,113)(72,121,136,117)(74,123,122,124)(78,126,99,116)(81,129,111,133)(93,135,134,130)(97,138,137,155)(101,120,145,146)(107,131,144,147)(125,153,140,149)(128,152,142,150)(141,157,158,151)(154,159,156,160), (1,2,5)(3,8,21)(4,11,15)(6,16,39)(7,18,34)(9,25,23)(10,28,59)(12,32,65)(13,20,38)(14,35,22)(17,42,40)(19,45,87)(24,52,76)(26,56,100)(27,48,72)(29,61,51)(30,54,49)(31,63,79)(33,69,67)(36,73,109)(37,41,66)(43,83,117)(44,60,110)(46,90,89)(47,92,88)(50,97,55)(53,91,57)(58,94,136)(62,113,121)(64,95,114)(68,77,84)(70,119,132)(71,82,80)(75,108,96)(78,125,93)(81,120,151)(85,112,102)(86,103,127)(98,138,135)(99,140,134)(101,141,144)(104,139,137)(105,126,153)(106,143,115)(107,145,158)(111,146,157)(116,149,118)(128,154,129)(130,148,155)(131,152,160)(133,142,156)(147,150,159));