Finite Group DatabasebetaFinite Group Databasebeta| order | 3024 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D3×GL(3,𝔽2) |
| Fitting | C32 ⋊ C2×GL(3,𝔽2) |
| socle | C32×GL(3,𝔽2) ⋊ C2 |
| derived | C3×GL(3,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×GL(3,𝔽2) |
| commutator, hypocenter | C3×GL(3,𝔽2) | C2×C3 |
| Frattini | C1 | G |
| Fitting | C32 | C2×GL(3,𝔽2) |
| socle | C32×GL(3,𝔽2) | C2 |
| cosocle | C3 | C2×C3×GL(3,𝔽2) |
| LCS | G ▷ C3×GL(3,𝔽2) | |
| UCS | C1 ◁ C3 | |
219 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 219 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) = 13 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(7,11,13)(8,9,12);(1,2,3)(4,6,5)(7,13)(9,10,12,11)54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×SO(3,𝔽7), #272×GL(3,𝔽2), C2×C3×D3×GL(3,𝔽2) |
| maximal normal subgroup of | C3×D3×SO(3,𝔽7), C2×C3×D3×GL(3,𝔽2) |
| maximal quotient of | C3×D3×SL(2,𝔽7), C2×C3×D3×GL(3,𝔽2) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(7,11,13)(8,9,12), (1,2,3)(4,6,5)(7,13)(9,10,12,11));