Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23 |
| direct product | C2×D4×3−1+2 |
| unambiguous semidirect product | C4⋊(C22×3−1+2) |
| unambiguous semidirect product | (C2×D4×C9)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C23×C32 |
| prime radical | C2×C32×D4 | C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(7,8,10,9,12,14,13,11,15);(1,2)(4,6)(8,11,12)(10,14,15);(1,2)(3,4)(5,6)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 405);