Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C3×D3×#120 (order 1152) · Inn(G) = D3 · Out(G) = C2×C3×#120 |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C4 |
| direct product | C4×C9×Dic3 |
| unambiguous semidirect product | C3⋊(C42×C9) |
| derived | C3 ⋊ C42×C9 |
| Sylow | C3×C9 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C9 | D3 |
| commutator, hypocenter | C3 | C42×C9 |
| Frattini | C22×C3 | C2×C3×D3 |
| Fitting | C2×C3×C4×C9 | C2 |
| socle, prime radical, cosocle | C22×C32 | C22×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C4×C9 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3)(4,11,10,6,5,12,9,8,7)(17,18,19,20);(1,2,3)(13,14,15,16)216 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×C9⋊C8, #11720258, Dic3⋊(C9⋊C8), … |
| maximal normal subgroup of | Dic3×C9⋊C8, #11720258, Dic3⋊(C9⋊C8), … |
| maximal quotient of | C4×C9×C3⋊C8, C9×#803, C8×C9×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 131);