Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C3⋊S4 |
| automorphism | Aut(G) = C22×((C3⋊S4)⋊C3)⋊C2 (order 1728) · Inn(G) = C3⋊S4 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| 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 | C3×A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊S4 |
| commutator, cosocle, hypocenter | C3×A4 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | D3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C2×C3 | |
208 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(4,5)(6,7,8)(9,10,12,11);(1,2)(3,4,5);(10,11,12)54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×(C3⋊S4)⋊C4, C3×#64207, C3×#64210, … |
| maximal normal subgroup of | C3×(C3⋊S4)⋊C4, C3×#64207, C3×#64210, … |
| maximal quotient of | C3×#64259, C3×C4×C3⋊S4, C3×C4⋊(C3⋊S4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 761);