Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×D3×B(4,𝔽2)⋊C3 |
| Fitting | C2×C3×B(4,𝔽2) ⋊ C2×C3 |
| derived | C3×H(5,𝔽2) ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×#120⋊C3 |
| commutator, cosocle, hypocenter | C3×H(5,𝔽2) | C23×C3 |
| Frattini | C23 | C22×D3×A4 |
| Fitting | C2×C3×B(4,𝔽2) | C2×C3 |
| socle | C22×C3 | C2×#120⋊C3 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C22 | |
2952 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 2952 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 48 |
A minimal faithful permutation representation — 3 generators on 13 points:
(4,5)(7,11,8)(10,13,12);(3,4,5)(6,13)(7,12,10,8)(9,11);(1,2)(7,10)(8,12)(11,13)96 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11714171, C2×D3×#73026, D3×#60399⋊C3, … |
| maximal normal subgroup of | D3×#11714171, C2×D3×#73026, D3×#60399⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5)(7,11,8)(10,13,12), (3,4,5)(6,13)(7,12,10,8)(9,11), (1,2)(7,10)(8,12)(11,13));