Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×S4×A5 |
| Fitting | C23 ⋊ D3×A5 |
| socle | C23×A5 ⋊ D3 |
| derived | A4×A5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | S4×A5 |
| commutator, hypocenter | A4×A5 | C22 |
| Frattini | C1 | G |
| Fitting | C23 | D3×A5 |
| socle | C23×A5 | D3 |
| cosocle | A4 | C22×A5 |
| LCS | G ▷ A4×A5 | |
| UCS | C1 ◁ C2 | |
623 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 623 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 11 points:
(3,5,6,4)(7,8,9,10,11);(1,2)(3,5)(7,10,8)50 conjugacy classes — view the full character table.
| maximal subgroup of | A5×GL(2,ℤ/4), S4×A5⋊C4, C2×A5×S5 |
| maximal normal subgroup of | A5×GL(2,ℤ/4), S4×A5⋊C4 |
| maximal quotient of | A5×GL(2,ℤ/4), S4×#6732, C2×GL(2,𝔽3)×A5, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,5,6,4)(7,8,9,10,11), (1,2)(3,5)(7,10,8));