Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×A4×S4 |
| Fitting, socle | C25×C5 ⋊ C3×D3 |
| derived | C22×A4 ⋊ C22×C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | A4×S4 |
| commutator, cosocle, hypocenter | C22×A4 | C22×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C25×C5 | C3×D3 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 ◁ C2×C5 | |
710 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(4,6,5)(7,8,10)(11,12,13,14,15);(3,6)(4,5)(8,9)200 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D5×A4×S4 |
| maximal normal subgroup of | C2×D5×A4×S4 |
| maximal quotient of | C5×#983⋊A4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,6,5)(7,8,10)(11,12,13,14,15), (3,6)(4,5)(8,9));