Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C3⋊D3×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | C3⋊D3×S4 |
| unambiguous semidirect product | (C3×S4)⋊D3 |
| Fitting, socle | C22×C32 ⋊ C2×D3 |
| derived | C32×A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×A4 | C22 |
| Fitting, socle | C22×C32 | C2×D3 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 | |
230 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,2,4,3)(8,9,10);(2,3,4)(6,7)(9,10);(5,6,7)30 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#157, D32×S4, C2×C3⋊D3×S4, … |
| maximal normal subgroup of | S4×#157, D32×S4, C2×C3⋊D3×S4, … |
| maximal quotient of | #11724176, #11724177, #11724178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 746);