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 | SD32 | A5 |
| hypocenter | SL(2,𝔽5) | D8 |
| LCS | G ▷ #91145 ▷ #6732 ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32 | |
182 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 182 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:
(1,82,65,78,2,18,58,72,81,29,21,84,39,96,32,20,8,89,51,31,23,54,41,77,34,75,57,64,53,86,5,56,42,92,10,19,59,90,94,30,7,95,70,50,12,47,26,40,52,61,49,55,35,91,63,4,28,87,80,9,6,83,73,93,11,46,85,69,24,60,22,25,14,76,33,48,27,71,79,62)(3,66,68,36,38,43,45,15,17,88,37,67,13,74,16,44)(97,126,175,186,123,115,129,188,110,152,118,182,120,160,181,171,107,173,169,150,102,127,176,185,124,116,159,192,131,180,145,125,148,168,122,114,108,174,166,178,103,158,190,167,155,144,106,172,138,121,101,156,149,165,153,142,130,189,137,179,119,105,147,183,104,100,128,187,135,151,117,157,177,136,154,143,98,146,161,191)(99,111,112,139,109,132,133,162,113,140,141,170,134,163,164,184);(1,111,34,115,59,173,17,158,53,160)(2,116,8,174,38,105,81,168,6,132)(3,126,24,165,7,140,11,144,27,189)(4,112,62,117,84,175,44,159,78,161)(5,139,63,142,85,187,45,182,80,183)(9,118,25,176,67,106,93,169,19,133)(10,143,26,188,68,125,94,186,21,162)(12,100,28,146,13,127,52,136,23,163)(14,152)(15,128,50,166,20,141,30,145,55,190)(16,156,51,185,22,170,32,171,57,192)(18,109,86,103,95,149,74,98,92,138)(29,119,54,177,88,107,96,135,46,113)(31,101,56,147,36,129,77,137,48,164)(33,114,58,172,37,157,79,167,49,184)(35,180)(39,151)(40,154)(41,179)(42,121)(43,108,76,110,47,134,60,97,82,120)(61,102,83,148,66,130,91,131,75,99)(64,181)(65,191)(69,153)(70,178)(71,155)(72,122)(73,150)(87,104)(89,124)(90,123)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240722);