Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C25 |
| direct product | S4×#5934 |
| Fitting | C22×H(5,𝔽2) ⋊ D32 |
| derived | A4×#997 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S4×#1020 |
| commutator, cosocle, hypocenter | A4×#997 | C22 |
| Fitting | C22×H(5,𝔽2) | D32 |
| socle | C23 | D3×#1020 |
| LCS | G ▷ A4×#997 | |
| UCS | C1 ◁ C2 | |
5328 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 5328 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,4,2,3)(6,11,9)(7,12,10);(2,3,4)(5,6,9,7)(8,10,12,11)65 conjugacy classes — view the full character table.
| maximal subgroup of | S4×AGL(3,𝔽2) |
| maximal quotient of | GL(2,𝔽3)×#5934 |
| automorphism group of | C22×#226, C22×GL(2,𝔽3), C22×SL(2,ℤ/4), … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,2,3)(6,11,9)(7,12,10), (2,3,4)(5,6,9,7)(8,10,12,11));