Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C3⋊(C3⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C33 |
| direct product | C2×C3⋊(C3⋊S4) |
| unambiguous semidirect product | (C2×C3)⋊(C3⋊S4) |
| unambiguous semidirect product | (C2×A4)⋊(C3⋊D3) |
| Fitting, socle | C23×C32 ⋊ D3 |
| derived | C32×A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C3⋊S4) |
| commutator, cosocle, hypocenter | C32×A4 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | D3 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(4,5)(7,8)(9,10,12,11);(1,2)(3,4,5);(6,7,8);(10,11,12)42 conjugacy classes — view the full character table.
| maximal subgroup of | #11724186, #11724187, #11724190, … |
| maximal normal subgroup of | #11724186, #11724187, #11724190, … |
| maximal quotient of | #11724463, C4×C3⋊(C3⋊S4), C4⋊(C3⋊(C3⋊S4)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 762);