Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | S4×C4⋊S4 |
| Fitting | C24×C4 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×S42 |
| commutator, cosocle | C2×A42 | C23 |
| Fitting | C24×C4 | D32 |
| socle | C25 | C2×D32 |
| hypercenter | C4 | S42 |
| hypocenter | A42 | C2×D4 |
| LCS | G ▷ C2×A42 ▷ A42 | |
| UCS | C1 ◁ C2 ◁ C4 | |
2560 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 2560 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(2,4)(5,6,8,7)(9,12,11,10);(6,7,8)(11,12);(1,2,3,4)70 conjugacy classes — view the full character table.
| maximal subgroup of | D4×S42, #72259050, #72259090 |
| maximal normal subgroup of | D4×S42, #72259050, #72259090 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4)(5,6,8,7)(9,12,11,10), (6,7,8)(11,12), (1,2,3,4));