Finite Group DatabasebetaFinite Group Databasebeta| order | 564 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C47 · Schur multiplier (H2) = C2 |
| direct product | A4×C47 |
| unambiguous semidirect product | C22⋊(C3×C47) |
| unambiguous semidirect product | (C22×C47)⋊C3 |
| Fitting, socle, Hall | C22×C47 ⋊ C3 |
| derived, Sylow | C22 ⋊ C3×C47 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C47 | A4 |
| commutator, cosocle, hypocenter | C22 | C3×C47 |
| Frattini | C1 | G |
| Fitting, socle | C22×C47 | C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C47 | |
10 conjugacy classes of subgroups.
| transitive degree | μt(G) = 188 |
188 conjugacy classes — view the full character table.
| derived subgroup of | A4⋊D47, S4×D47, S4×C47⋊C23 |
| maximal subgroup of | S4×C47, A4⋊D47, A4×D47, … |
| maximal normal subgroup of | S4×C47, A4⋊D47, A4×D47, … |
| maximal quotient of | SL(2,𝔽3)×C47, C2×A4×C47, C22⋊C9×C47, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(564, 6);