Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D32×A4 |
| automorphism | Aut(G) = S4×GO+(4,𝔽2) (order 1728) · Inn(G) = G · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | D32×A4 |
| unambiguous semidirect product | (C3×D3)⋊(C2×A4) |
| unambiguous semidirect product | (C22×D32)⋊C3 |
| Fitting, socle, derived | C22×C32 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C32 | C22×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 | |
188 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 30 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:
(2,3,4)(5,6,7)(9,10);(1,4)(2,3)(6,7)(8,9,10)36 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊S4, A4×GO+(4,𝔽2), D32×S4, … |
| maximal normal subgroup of | D32⋊S4, A4×GO+(4,𝔽2), D32×S4, … |
| maximal quotient of | D3×C3⋊(#41⋊C3), #64319⋊C3, #64316⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 749);