Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×S42 |
| Fitting | C24×C4 ⋊ D32 |
| derived | A42 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | S42 |
| commutator, hypocenter | A42 | C22×C4 |
| Frattini | C2 | C2×S42 |
| Fitting | C24×C4 | D32 |
| socle | C25 | C2×D32 |
| prime radical | C2×S42 | C2 |
| cosocle | C2×A42 | C23 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C4 | |
2552 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 2552 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 36 |
A minimal faithful permutation representation — 3 generators on 12 points:
(5,8,6,7)(9,11,12,10);(1,3,4,2)(6,7,8);(1,2,4,3)(11,12)100 conjugacy classes — view the full character table.
| maximal subgroup of | #72228310, AGL(1,𝔽5)×S42, C4×#10596770⋊C2, … |
| maximal normal subgroup of | #72228310, C4×#10596770⋊C2, D4×S42, … |
| maximal quotient of | AGL(1,𝔽5)×S42 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,8,6,7)(9,11,12,10), (1,3,4,2)(6,7,8), (1,2,4,3)(11,12));