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 | #11751487 |
| commutator, cosocle | #91145 | C22 |
| Frattini | C8 | C2×S5 |
| Fitting, hypercenter | C16 | S5 |
| prime radical | #6732 | D4 |
| hypocenter | SL(2,𝔽5) | D8 |
| LCS | G ▷ #91145 ▷ #6732 ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
142 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 142 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:
(1,66,33,77,28,47,16,30,24,54,21,67,2,92,85,4,17,61,79,56,23,43,32,50,27,46,68,9,81,95,5,44,34,91,58,48,13,60,51,55,7,88,10,93,8,18,45,62,53,83,49,36,12,76,57,20,3,29,94,25,6,74,63,78,59,75,37,31,52,82,22,15,11,96,26,19,38,86,80,84)(14,89,70,69,39,90,73,72,42,87,65,64,35,71,41,40)(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);(1,97,7,103)(2,98,12,108)(3,99,17,113)(4,100,20,116)(5,101,22,118)(6,102,23,119)(8,104,28,124)(9,105,31,127)(10,106,33,129)(11,107,34,130)(13,109,38,134)(14,110,42,138)(15,111,44,140)(16,112,45,141)(18,114,47,143)(19,115,48,144)(21,117,49,145)(24,120,53,149)(25,121,56,152)(26,122,58,154)(27,123,59,155)(29,125,61,157)(30,126,62,158)(32,128,63,159)(35,131,39,135)(36,132,67,163)(37,133,68,164)(40,136,72,168)(41,137,73,169)(43,139,74,170)(46,142,75,171)(50,146,78,174)(51,147,80,176)(52,148,81,177)(54,150,83,179)(55,151,84,180)(57,153,85,181)(60,156,86,182)(64,160,69,165)(65,161,70,166)(66,162,88,184)(71,167,90,186)(76,172,92,188)(77,173,93,189)(79,175,94,190)(82,178,95,191)(87,183,89,185)(91,187,96,192)45 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240716);