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 | D8×A5 |
| commutator | #91145 | C22 |
| Frattini, cosocle | C8 | C22×A5 |
| Fitting, hypercenter | Q32 | A5 |
| hypocenter | SL(2,𝔽5) | D8 |
| LCS | G ▷ #91145 ▷ #6732 ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32 | |
183 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 15 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:
(1,4,5,18,6,19,21,46,7,20,22,47,23,48,49,75)(2,55,41,74,53,30,58,87,3,93,33,95,39,36,51,86,27,40,45,91,11,56,65,88,81,31,85,90,13,50,63,54,14,44,79,29,28,64,68,92,12,84,73,43,24,62,26,89,17,77,10,82,35,67,80,60,59,72,16,96,34,25,70,66,52,9,57,71,38,78,32,83,42,15,94,61,8,69,37,76)(97,182,169,151,134,143,129,160,104,162,117,126,110,179,112,100,98,186,153,163,119,157,161,121,113,142,128,136,155,139,101,105,135,178,164,144,130,183,122,140,103,156,137,180,109,114,106,165,124,184,145,158,138,150,141,116,108,167,181,132,102,125,166,152,99,171,159,168,123,170,118,127,131,191,133,115,107,185,154,111)(120,192,190,189,177,188,176,174,149,187,175,173,148,172,147,146);(1,134,11,103,59,120,3,98,53,110,7,109,34,97,27,149,17,108,24,138)(2,102,28,177,38,130,52,135,6,99,12,119,8,148,13,107,81,131,23,113)(4,163,30,116,84,146,15,105,78,136,20,132,62,100,55,174,44,127,50,168)(5,164,32,118,85,147,16,106,80,137,22,133,63,101,57,176,45,129,51,169)(9,115,56,189,67,158,77,165,19,111,31,144,25,173,36,126,93,160,48,140)(10,117,58,190,68,159,79,166,21,112,33,145,26,175,37,128,94,161,49,141)(14,123,42,155)(18,184,60,143,95,172,43,125,92,167,47,162,86,114,82,188,74,157,76,186)(29,142,83,192,88,182,91,185,46,139,61,171,54,187,66,156,96,183,75,170)(35,124,39,104)(40,151,72,180)(41,153,73,181)(64,152,69,121)(65,154,70,122)(71,178,90,191)(87,179,89,150)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240721);