Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini | C26 | #11757162 |
| Fitting, cosocle | C2×#6342371 | A5 |
| socle | C24 | #72209604 |
| LCS | G | |
| UCS | C1 | |
At least 19058 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has at least 19058 conjugacy classes of subgroups — too many to draw. Only its 9 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 — 2 generators on 160 points:
(1,25,111,74,101)(2,46,96,65,121)(3,31,117,92,138)(4,83,73,156,59)(5,60,134,75,119)(6,57,122,148,100)(7,45,116,15,136)(8,72,58,146,19)(9,42,105,27,124)(10,43,139,64,80)(11,132,107,86,127)(12,143,71,150,24)(13,33,118,62,154)(14,133,63,120,32)(16,108,103,79,129)(17,88,85,151,90)(18,44,110,160,66)(20,93,147,89,144)(21,36,130,135,128)(22,67,123,112,81)(23,76,109,87,137)(26,70,142,50,145)(28,39,158,97,114)(29,115,98,155,51)(30,91,77,141,126)(34,38,152,106,113)(35,125,47,102,55)(37,149,52,54,140)(40,99,61,78,131)(41,82,48,94,104)(49,84,69,159,56)(53,68,157,95,153);(1,2,6)(3,11,34)(4,14,29)(5,10,21)(7,23,26)(8,18,43)(9,30,20)(12,36,45)(13,40,28)(15,44,80)(16,24,51)(17,46,31)(19,48,53)(22,52,32)(27,61,100)(33,49,42)(35,68,91)(37,72,82)(38,57,76)(39,67,70)(41,77,119)(47,83,121)(50,88,124)(54,60,93)(58,96,136)(59,66,90)(62,105,145)(63,107,81)(64,103,126)(65,104,128)(69,109,150)(71,111,138)(73,116,153)(74,117,129)(75,118,127)(78,122,144)(79,123,149)(84,120,87)(85,125,113)(86,130,101)(89,115,131)(92,133,102)(94,132,99)(95,134,146)(97,140,155)(98,139,137)(106,147,154)(110,142,114)(112,152,160)(135,143,159)(141,158,156)(148,157,151)32 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((1,25,111,74,101)(2,46,96,65,121)(3,31,117,92,138)(4,83,73,156,59)(5,60,134,75,119)(6,57,122,148,100)(7,45,116,15,136)(8,72,58,146,19)(9,42,105,27,124)(10,43,139,64,80)(11,132,107,86,127)(12,143,71,150,24)(13,33,118,62,154)(14,133,63,120,32)(16,108,103,79,129)(17,88,85,151,90)(18,44,110,160,66)(20,93,147,89,144)(21,36,130,135,128)(22,67,123,112,81)(23,76,109,87,137)(26,70,142,50,145)(28,39,158,97,114)(29,115,98,155,51)(30,91,77,141,126)(34,38,152,106,113)(35,125,47,102,55)(37,149,52,54,140)(40,99,61,78,131)(41,82,48,94,104)(49,84,69,159,56)(53,68,157,95,153), (1,2,6)(3,11,34)(4,14,29)(5,10,21)(7,23,26)(8,18,43)(9,30,20)(12,36,45)(13,40,28)(15,44,80)(16,24,51)(17,46,31)(19,48,53)(22,52,32)(27,61,100)(33,49,42)(35,68,91)(37,72,82)(38,57,76)(39,67,70)(41,77,119)(47,83,121)(50,88,124)(54,60,93)(58,96,136)(59,66,90)(62,105,145)(63,107,81)(64,103,126)(65,104,128)(69,109,150)(71,111,138)(73,116,153)(74,117,129)(75,118,127)(78,122,144)(79,123,149)(84,120,87)(85,125,113)(86,130,101)(89,115,131)(92,133,102)(94,132,99)(95,134,146)(97,140,155)(98,139,137)(106,147,154)(110,142,114)(112,152,160)(135,143,159)(141,158,156)(148,157,151));