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 | D8 | S5 |
| prime radical | #11751545 | C2 |
| hypocenter | SL(2,𝔽5) | D8 |
| LCS | G ▷ #91145 ▷ #6732 ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
159 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 159 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:
(1,112,7,137,30,140,34,143,9,138)(2,139,17,111,38,127,25,129,18,133)(3,104)(4,113,19,99,43,131,44,103,22,108)(5,128,20,109,28,141,35,115,10,125)(6,100,21,122,12,144,46,132,24,136)(8,124,32,102,47,123,40,130,33,134)(11,118)(13,105)(14,119)(15,97,36,107,42,142,45,116,23,121)(16,101,37,110,29,114,48,117,39,126)(26,106)(27,135)(31,120)(41,98)(49,160,62,187,72,189,83,170,86,152)(50,192,94,175,55,166,53,176,75,186)(51,173,57,177,66,185,68,167,54,148)(52,161,79,172,87,190,93,158,95,153)(56,162,96,171,67,154,63,145,89,156)(58,188,82,157,69,183,65,159,59,184)(60,191)(61,169,70,178,81,147,84,168,64,149)(71,179,92,155,85,146,80,150,74,174)(73,163)(76,181)(77,151)(78,180)(88,164)(90,182)(91,165);(1,166,6,183)(2,175,10,157)(3,176,14,159)(4,154,16,146)(5,167,17,152)(7,184,21,186)(8,171,23,155)(9,185,24,170)(11,148,27,160)(12,177,30,189)(13,145,31,150)(15,168,32,153)(18,180,35,191)(19,174,37,156)(20,187,38,173)(22,147,39,158)(25,188,28,192)(26,149,41,161)(29,178,43,190)(33,165,45,151)(34,181,46,163)(36,172,47,169)(40,179,42,162)(44,182,48,164)(49,121,54,124)(50,142,58,130)(51,107,62,123)(52,136,64,138)(53,126,65,108)(55,134,69,116)(56,144,71,140)(57,114,72,131)(59,110,75,99)(60,101,78,113)(61,122,79,137)(63,139,80,125)(66,120,83,105)(67,143,85,132)(68,103,86,117)(70,129,87,141)(73,102,76,97)(74,127,89,109)(77,111,91,128)(81,135,93,118)(82,98,94,106)(84,115,95,133)(88,112,90,100)(92,104,96,119)36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240714);