Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | #91883 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | A4×S5 |
| commutator, hypocenter | #91883 | C3×C4 |
| Fitting | #41 | C3×S5 |
| socle | C2 | A4×A5⋊C4 |
| prime radical | #72276703 | C2 |
| cosocle | #11756696 | C2×C3 |
| LCS | G ▷ #91883 | |
| UCS | C1 ◁ C4 | |
349 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 96 points:
(1,54,66,12,50,59,90,20,88,13,71,22,2,68,65,27,57,58,85,34,87,15,44,26)(3,56,73,45,6,18,42,23,69,40,17,92,7,49,72,46,10,21,41,25,63,39,32,94)(4,81,61,43,78,52,96,31,86,48,84,33,8,79,62,67,77,74,95,16,91,55,83,19)(5,75,38,35,93,30,80,28,51,64,70,11,9,60,37,24,89,14,82,36,47,53,76,29);(1,5,2,9)(3,17,15,56)(4,18,8,21)(6,27,25,41)(7,32,13,49)(10,12,23,42)(11,24,38,54)(14,53,51,20)(16,58,55,88)(19,63,62,75)(22,67,66,79)(26,43,65,81)(28,52,73,91)(29,35,37,68)(30,64,47,34)(31,59,48,87)(33,69,61,60)(36,74,72,86)(39,71)(40,44)(45,57)(46,50)(70,77)(76,78)(80,85,82,90)(83,89)(84,93)(92,95,94,96)86 conjugacy classes — view the full character table.
| maximal quotient of | SL(2,𝔽3)×GL(2,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,54,66,12,50,59,90,20,88,13,71,22,2,68,65,27,57,58,85,34,87,15,44,26)(3,56,73,45,6,18,42,23,69,40,17,92,7,49,72,46,10,21,41,25,63,39,32,94)(4,81,61,43,78,52,96,31,86,48,84,33,8,79,62,67,77,74,95,16,91,55,83,19)(5,75,38,35,93,30,80,28,51,64,70,11,9,60,37,24,89,14,82,36,47,53,76,29), (1,5,2,9)(3,17,15,56)(4,18,8,21)(6,27,25,41)(7,32,13,49)(10,12,23,42)(11,24,38,54)(14,53,51,20)(16,58,55,88)(19,63,62,75)(22,67,66,79)(26,43,65,81)(28,52,73,91)(29,35,37,68)(30,64,47,34)(31,59,48,87)(33,69,61,60)(36,74,72,86)(39,71)(40,44)(45,57)(46,50)(70,77)(76,78)(80,85,82,90)(83,89)(84,93)(92,95,94,96));