Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | S4×S5 |
| Fitting | C22 ⋊ D3×S5 |
| socle | C22×A5 ⋊ C2×D3 |
| derived | A4×A5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A4×A5 | C22 |
| Fitting | C22 | D3×S5 |
| socle | C22×A5 | C2×D3 |
| LCS | G ▷ A4×A5 | |
| UCS | C1 | |
467 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 467 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2,4,3)(5,6,7,8,9);(2,4)(5,8,7)(6,9)35 conjugacy classes — view the full character table.
| maximal subgroup of | S52 |
| maximal quotient of | S4×A5⋊C4 |
| automorphism group of | C22×S5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,3)(5,6,7,8,9), (2,4)(5,8,7)(6,9));