Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D9⋊A4 |
| automorphism | Aut(G) = C2×A4×AGL(1,ℤ/9) (order 1296) · Inn(G) = D9⋊A4 · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×D9⋊A4 |
| unambiguous semidirect product | D9⋊(C2×A4) |
| unambiguous semidirect product | (C2×D9)⋊A4 |
| unambiguous semidirect product | (C9⋊A4)⋊C22 |
| unambiguous semidirect product | (C2×C9⋊A4)⋊C2 |
| Fitting | C23×C9 ⋊ C2×C3 |
| derived | C22×C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9⋊A4 |
| commutator, cosocle, hypocenter | C22×C9 | C22×C3 |
| Frattini | C3 | C2×D3×A4 |
| Fitting | C23×C9 | C2×C3 |
| socle | C23×C3 | C3×D3 |
| LCS | G ▷ C22×C9 | |
| UCS | C1 ◁ C2 | |
121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(4,6,5)(8,12,15,10,11,14)(9,13);(1,2)(3,6)(4,5)(7,11,14,13,8,10,9,15,12)32 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D9⋊A4, C4⋊(D9⋊A4), #611⋊A4, … |
| maximal normal subgroup of | C4×D9⋊A4, C4⋊(D9⋊A4), #611⋊A4, … |
| maximal quotient of | Dic18⋊A4, C4×D9⋊A4, C4⋊(D9⋊A4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 539);