Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×A4⋊S4 |
| unambiguous semidirect product | (C2×A4)⋊S4 |
| Fitting, socle | C25 ⋊ C3⋊D3 |
| derived | A42 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4⋊S4 |
| commutator, cosocle, hypocenter | A42 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C25 | C3⋊D3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2 | |
337 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 18 |
28 conjugacy classes — view the full character table.
| maximal subgroup of | #11923458, #11923460, #11923469, … |
| maximal normal subgroup of | #11923458, #11923460, #11923469, … |
| maximal quotient of | #11923507, C4×A4⋊S4, C4⋊(A4⋊S4), … |
| holomorph of | C2×A4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8657);