Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | D3×AGL(1,𝔽5)×S4 |
| Fitting, socle | C22×C3×C5 ⋊ C2×C4×D3 |
| derived | C3×C5×A4 ⋊ C22×C4 |
| Sylow | C5 ⋊ C4×D3×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C3×C5×A4 | C22×C4 |
| Fitting, socle | C22×C3×C5 | C2×C4×D3 |
| prime radical | D3×D5×S4 | C2 |
| cosocle | C3×D5×A4 | C23 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 | |
964 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,4,3)(6,7,8,9)(10,11,12);(5,6,9,7,8)(11,12);(2,4)75 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)(6,7,8,9)(10,11,12), (5,6,9,7,8)(11,12), (2,4));