Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C3 |
| direct product | C23×C32×D3 |
| unambiguous semidirect product | C3⋊(C24×C32) |
| unambiguous semidirect product | (C2×C3)⋊(C23×C32) |
| Fitting, socle | C23×C33 ⋊ C2 |
| derived | C3 ⋊ C24×C32 |
| Sylow | C33 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C32 | D3 |
| commutator, cosocle, hypocenter | C3 | C24×C32 |
| Frattini | C1 | G |
| Fitting, socle | C23×C33 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23×C32 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 15 points:
(8,9)(10,11,12);(5,6)(7,8,9)(13,14,15);(3,4);(1,2)216 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×(C2×D3)⋊Dic3, C3×D3×#217, C3×#63976, … |
| maximal normal subgroup of | C2×C3×(C2×D3)⋊Dic3, C3×D3×#217, C3×#63976, … |
| maximal quotient of | C22×C32×Dic6, C22×C32×C4×D3, C22×C32×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 772);