Finite Group DatabasebetaFinite Group Databasebeta| order | 3024 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C3 |
| direct product | C23×C3⋊D3×ΓL(1,𝔽8) |
| Fitting, socle | C23×C32×C7 ⋊ C2×C3 |
| derived | C32×C7 ⋊ C24×C3 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C24 |
| Sylow | C7 ⋊ C23×C3×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊D3×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C32×C7 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32×C7 | C2×C3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C23 | |
1864 conjugacy classes of subgroups; only the 441 normal subgroups are shown.
This group has 1864 conjugacy classes of subgroups — too many to draw. Only its 441 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 4 generators on 19 points:
(5,6)(7,8,9)(14,15,16)(17,19,18);(8,9)(11,12)(13,14,16,18,15,17,19);(3,4)(10,11,12);(1,2)240 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((5,6)(7,8,9)(14,15,16)(17,19,18), (8,9)(11,12)(13,14,16,18,15,17,19), (3,4)(10,11,12), (1,2));