Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C3×C4 |
| direct product | C42×A4⋊S4 |
| Fitting | C24×C42 ⋊ C3⋊D3 |
| derived | A42 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | A4⋊S4 |
| commutator, hypocenter | A42 | C2×C42 |
| Frattini | C22 | C22×A4⋊S4 |
| Fitting | C24×C42 | C3⋊D3 |
| socle | C26 | C22×C3⋊D3 |
| prime radical | C22×A4⋊S4 | C22 |
| cosocle | C22×A42 | C23 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C42 | |
6988 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 6988 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 16 points:
(5,6,7,8)(9,10,12)(13,16,15);(10,11)(15,16);(1,2,3,4)(14,16,15)224 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((5,6,7,8)(9,10,12)(13,16,15), (10,11)(15,16), (1,2,3,4)(14,16,15));