Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×S4×GL(2,𝔽3) |
| Fitting | C23×Q8 ⋊ D32 |
| derived | A4×SL(2,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | S42 |
| commutator, cosocle, hypocenter | A4×SL(2,𝔽3) | C23 |
| Frattini | C2 | C2×S42 |
| Fitting | C23×Q8 | D32 |
| socle | C24 | D3×S4 |
| LCS | G ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
2311 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 2311 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,4,6)(7,13,14)(8,12,10);(8,11)(9,13)(12,14);(4,5)80 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((1,2)(3,4,6)(7,13,14)(8,12,10), (8,11)(9,13)(12,14), (4,5));