Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3⋊D3×GL(3,𝔽2) |
| Fitting | C2×C32 ⋊ C2×GL(3,𝔽2) |
| socle | C2×C32×GL(3,𝔽2) ⋊ C2 |
| derived | C32×GL(3,𝔽2) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊D3×GL(3,𝔽2) |
| commutator, hypocenter | C32×GL(3,𝔽2) | C22 |
| Frattini | C1 | G |
| Fitting | C2×C32 | C2×GL(3,𝔽2) |
| socle | C2×C32×GL(3,𝔽2) | C2 |
| cosocle | C32 | C22×GL(3,𝔽2) |
| LCS | G ▷ C32×GL(3,𝔽2) | |
| UCS | C1 ◁ C2 | |
994 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 3 generators on 15 points:
(3,4)(6,7)(9,10,12)(11,13,15);(1,2)(4,5)(7,8)(9,11,14,15)(10,12);(6,7,8)72 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(6,7)(9,10,12)(11,13,15), (1,2)(4,5)(7,8)(9,11,14,15)(10,12), (6,7,8));