Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C23×C3×S4 |
| unambiguous semidirect product | A4⋊(C24×C3) |
| unambiguous semidirect product | (C2×A4)⋊(C23×C3) |
| Fitting, socle | C25×C3 ⋊ D3 |
| derived | A4 ⋊ C24×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | S4 |
| commutator, cosocle, hypocenter | A4 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C25×C3 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C23×C3 | |
1434 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8668);