Finite Group DatabasebetaFinite Group Databasebeta| order | 528 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C22 |
| direct product | C2×C11×S4 |
| unambiguous semidirect product | A4⋊(C22×C11) |
| unambiguous semidirect product | (C2×A4)⋊(C2×C11) |
| Fitting, socle | C23×C11 ⋊ D3 |
| derived | A4 ⋊ C22×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C11 | S4 |
| commutator, cosocle, hypocenter | A4 | C22×C11 |
| Frattini | C1 | G |
| Fitting, socle | C23×C11 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C11 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 66 |
110 conjugacy classes — view the full character table.
| maximal subgroup of | S4×Dic11, #11761328, #11761330, … |
| maximal normal subgroup of | S4×Dic11, #11761328, #11761330, … |
| maximal quotient of | C11×A4⋊Q8, C4×C11×S4, C11×C4⋊S4, … |
| automorphism group of | Q8×C23, A4×C23 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(528, 161);