Finite Group DatabasebetaFinite Group Databasebeta| order | 528 |
| isoclinism | isoclinic class of C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C11 · Schur multiplier (H2) = C4 |
| direct product | C11×C42⋊C3 |
| unambiguous semidirect product | C42⋊(C3×C11) |
| unambiguous semidirect product | (C42×C11)⋊C3 |
| Fitting, Hall | C42×C11 ⋊ C3 |
| derived, Sylow | C42 ⋊ C3×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C11 | C42⋊C3 |
| commutator, cosocle, hypocenter | C42 | C3×C11 |
| Frattini | C22 | C11×A4 |
| Fitting | C42×C11 | C3 |
| socle | C22×C11 | A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C11 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 132 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16)(14,18,23)(15,19,22)(17,21,20);(12,14,15,20)(13,19)(16,23,22,21)(17,18)88 conjugacy classes — view the full character table.
| derived subgroup of | C11⋊#857 |
| maximal subgroup of | C11×#857, C11⋊#857, D11×C42⋊C3, … |
| maximal normal subgroup of | C11×#857, C11⋊#857, D11×C42⋊C3, … |
| maximal quotient of | C11×#95⋊C3, C2×C11×C42⋊C3, C11×C42⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(528, 30);