Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C8×A4⋊S4 |
| Fitting | C24×C8 ⋊ C3⋊D3 |
| derived | A42 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | A4⋊S4 |
| commutator, hypocenter | A42 | C2×C8 |
| Frattini | C4 | C2×A4⋊S4 |
| Fitting | C24×C8 | C3⋊D3 |
| socle | C25 | C4×C3⋊D3 |
| prime radical | C2×A4⋊S4 | C4 |
| cosocle | C4×A42 | C22 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C8 | |
859 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 859 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 72 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,4,3,2)(5,8,6,7);(6,7,8)(9,10,11,13,12,14,15,16);(2,3,4)(6,7,8)112 conjugacy classes — view the full character table.
| maximal subgroup of | M16×A4⋊S4, SD16×A4⋊S4, C2×C8×A4⋊S4, … |
| maximal normal subgroup of | M16×A4⋊S4, SD16×A4⋊S4, C2×C8×A4⋊S4, … |
| maximal quotient of | C2×C8×A4⋊S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3,2)(5,8,6,7), (6,7,8)(9,10,11,13,12,14,15,16), (2,3,4)(6,7,8));