Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of A4×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×A4×S4 |
| unambiguous semidirect product | A4⋊(C22×A4) |
| unambiguous semidirect product | (C2×A4)⋊(C2×A4) |
| unambiguous semidirect product | (C23×S4)⋊C3 |
| Fitting, socle | C25 ⋊ C3×D3 |
| derived | C22×A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×S4 |
| commutator, cosocle, hypocenter | C22×A4 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C25 | C3×D3 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 ◁ C2 | |
355 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 18 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | S4×SL(2,ℤ/4), #11923459, #11923460, … |
| maximal normal subgroup of | S4×SL(2,ℤ/4), #11923459, #11923460, … |
| maximal quotient of | A4×A4⋊Q8, C4×A4×S4, A4×C4⋊S4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8655);