Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C2×C4×#272 (order 432) · Inn(G) = C32 · Out(G) = C2×C4×D3 |
| homology | Abelianization (H1) = C32×C16 · Schur multiplier (H2) = C1 |
| direct product | C16×3−1+2 |
| unambiguous semidirect product | (C9×C16)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C16 | C32 |
| commutator | C3 | C32×C16 |
| Frattini, cosocle | C3×C8 | C2×C32 |
| socle | C2×C3 | C32×C8 |
| prime radical | C2×C32 | C3×C8 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C16 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,15,13,16,18,21,14,17,19,22,20,23,24,25);(2,5,6)(4,8,9)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 36);