Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D3×S4 |
| automorphism | Aut(G) = C2×D3×S4 (order 288) · Inn(G) = D3×S4 · Out(G) = C2 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×D3×S4 |
| unambiguous semidirect product | (C3×S4)⋊(C2×C3) |
| unambiguous semidirect product | (D3×A4)⋊(C2×C3) |
| Fitting, socle | C22×C32 ⋊ C2×D3 |
| derived | C3×A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×S4 |
| commutator, cosocle, hypocenter | C3×A4 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32 | C2×D3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C3 | |
163 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 24 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2,4,3)(5,6,7)(8,9,10);(2,3,4)(9,10)45 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 745);