Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11751518 |
| commutator, cosocle | #91145 | C22 |
| Frattini | C8 | C2×S5 |
| Fitting, hypercenter | Q16 | S5 |
| prime radical | #11751549 | C2 |
| hypocenter | SL(2,𝔽5) | D8 |
| LCS | G ▷ #91145 ▷ #6732 ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
160 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 192 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 192 points:
(1,125)(2,112)(3,135)(4,121)(5,136)(6,139)(7,115)(8,113)(9,128)(10,100)(11,104)(12,129)(13,98)(14,118)(15,126)(16,124)(17,138)(18,137)(19,116)(20,132)(21,133)(22,97)(23,101)(24,111)(25,140)(26,105)(27,119)(28,144)(29,130)(30,141)(31,106)(32,108)(33,99)(34,109)(35,122)(36,117)(37,134)(38,143)(39,102)(40,131)(41,120)(42,114)(43,142)(44,107)(45,110)(46,127)(47,103)(48,123)(49,166)(50,177)(51,157)(52,154)(53,167)(54,183)(55,148)(56,178)(57,188)(58,189)(59,170)(60,181)(61,155)(62,175)(63,168)(64,146)(65,152)(66,184)(67,149)(68,159)(69,160)(70,179)(71,190)(72,192)(73,191)(74,158)(75,185)(76,180)(77,182)(78,163)(79,171)(80,153)(81,174)(82,187)(83,186)(84,150)(85,161)(86,176)(87,162)(88,151)(89,147)(90,165)(91,164)(92,172)(93,156)(94,173)(95,145)(96,169);(1,73,47,96,5,78,19,81,6,76,36,92,17,60,37,93)(2,69,8,85,9,86,22,95,10,55,23,67,24,68,39,84)(3,75,33,64,11,51,44,80,14,59,45,52,27,62,48,63)(4,88,7,66,15,91,20,82,16,90,21,83,32,77,38,94)(12,58,43,56,25,72,42,70,30,50,29,71,28,57,40,87)(13,89,34,65,26,61,35,49,31,74,46,53,41,79,18,54)(97,192,104,171,113,189,135,158,102,188,119,155,101,177,118,147)(98,175,112,179,120,170,111,178,106,157,100,162,105,185,128,190)(99,186,122,169,123,187,109,156,110,184,137,172,107,173,127,174)(103,159,129,151,134,148,144,182,117,176,141,165,116,160,140,164)(108,167,138,154,124,166,139,153,126,152,136,146,121,183,125,168)(114,191,133,145,142,181,132,161,131,180,115,150,130,163,143,149)36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240713);