Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | D3×C22⋊S5 |
| Fitting | C22×C3 ⋊ C2×S5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×S5 |
| commutator, socle, cosocle | C2×C3×A5 | C23 |
| Fitting | C22×C3 | C2×S5 |
| hypercenter | C22 | D3×S5 |
| hypocenter | C3×A5 | C2×D4 |
| LCS | G ▷ C2×C3×A5 ▷ C3×A5 | |
| UCS | C1 ◁ C2 ◁ C22 | |
945 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 945 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3)(4,7,5,6)(8,11,12,9);(1,2)(4,6,5,7)(8,11);(1,2)(6,7)(8,12,9,10)57 conjugacy classes — view the full character table.
| maximal subgroup of | D3×A4⋊S5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,7,5,6)(8,11,12,9), (1,2)(4,6,5,7)(8,11), (1,2)(6,7)(8,12,9,10));