Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | Aut(G) = #120×#272 (order 1728) · Inn(G) = C22×C32 · Out(G) = D3×D4 |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C22 |
| direct product | C22⋊C4×3−1+2 |
| unambiguous semidirect product | (C9×C22⋊C4)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C22×C3 | C22×C32 |
| commutator | C2×C3 | C2×C32×C4 |
| prime radical | C23×C32 | C2×C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,13)(14,15)(16,17);(2,5,6)(4,8,9)(15,17)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 205);