Finite Group DatabasebetaFinite Group Databasebeta| order | 2448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×S4×D17 |
| Fitting, socle | C22×C3×C17 ⋊ C2×D3 |
| derived | A4×C17 ⋊ C22×C3 |
| Sylow | C17 ⋊ C2×C3×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | S4×D17 |
| commutator, cosocle, hypocenter | A4×C17 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C17 | C2×D3 |
| LCS | G ▷ A4×C17 | |
| UCS | C1 ◁ C3 | |
140 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 204 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,4,3)(5,6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24);(2,4)(8,9,11,13,15,17,19,21,23,24,22,20,18,16,14,12,10)150 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((1,4,3)(5,6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (2,4)(8,9,11,13,15,17,19,21,23,24,22,20,18,16,14,12,10));