Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C3 |
| direct product | #157×GL(3,𝔽2) |
| Fitting | C32 ⋊ C4×GL(3,𝔽2) |
| socle, derived | C32×GL(3,𝔽2) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | C32×GL(3,𝔽2) | C4 |
| Fitting | C32 | C4×GL(3,𝔽2) |
| prime radical | C3⋊D3×GL(3,𝔽2) | C2 |
| cosocle | C3⋊D3 | C2×GL(3,𝔽2) |
| LCS | G ▷ C32×GL(3,𝔽2) | |
| UCS | C1 | |
240 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 2 generators on 13 points:
(4,6,5)(9,10)(12,13);(1,6)(2,4,3,5)(7,8,9,11,12,13,10)36 conjugacy classes — view the full character table.
| maximal subgroup of | GL(3,𝔽2)×A6 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,6,5)(9,10)(12,13), (1,6)(2,4,3,5)(7,8,9,11,12,13,10));