Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D9×S4 |
| automorphism | Aut(G) = S4×AGL(1,ℤ/9) (order 1296) · Inn(G) = G · Out(G) = C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | D9×S4 |
| unambiguous semidirect product | (C9×S4)⋊C2 |
| unambiguous semidirect product | (C9⋊S4)⋊C2 |
| unambiguous semidirect product | (A4×D9)⋊C2 |
| Fitting | C22×C9 ⋊ C2×D3 |
| derived | C9×A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C9×A4 | C22 |
| Frattini | C3 | D3×S4 |
| Fitting | C22×C9 | C2×D3 |
| socle | C22×C3 | D32 |
| LCS | G ▷ C9×A4 | |
| UCS | C1 | |
107 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 107 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,4,3)(5,10,8,7,12,9,11,6,13);(2,3,4)(6,8)(7,11)(9,12)(10,13)30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D9×S4, S4×D27, C3×D9×S4, … |
| maximal normal subgroup of | C2×D9×S4, C3×D9×S4, S4×AGL(1,ℤ/9), … |
| maximal quotient of | D9×#226, #11722231, #11722232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 521);