Finite Group DatabasebetaFinite Group Databasebeta| order | 528 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C4×C11×A4 |
| unambiguous semidirect product | (C22×C4)⋊(C3×C11) |
| unambiguous semidirect product | (C22×C4×C11)⋊C3 |
| Fitting, Hall | C22×C4×C11 ⋊ C3 |
| derived | C22 ⋊ C3×C4×C11 |
| Sylow | C22×C4 ⋊ C3×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C11 | A4 |
| commutator, hypocenter | C22 | C3×C4×C11 |
| Frattini | C2 | C2×C11×A4 |
| Fitting | C22×C4×C11 | C3 |
| socle | C23×C11 | C2×C3 |
| prime radical | C2×C11×A4 | C2 |
| cosocle | C23 | C2×C3×C11 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C4×C11 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 132 |
176 conjugacy classes — view the full character table.
| maximal subgroup of | C11×A4⋊C8, C11⋊(A4⋊C8), A4×C11⋊C8, … |
| maximal normal subgroup of | C11×A4⋊C8, C11⋊(A4⋊C8), A4×C11⋊C8, … |
| maximal quotient of | C4×C11×SL(2,𝔽3), C8×C11×A4, C11×#131⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(528, 123);