Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×C9×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C9 | C22 |
| commutator | C2 | C23×C3×C9 |
| Frattini, cosocle | C2×C3 | C23×C32 |
| socle | C2×C32 | C23×C3 |
| prime radical | C32×#41 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C9 ◁ G | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11);(2,5)(6,8)(12,13,15,14,16,18,17,19,20);(1,3,4,7)(2,6,5,8)270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 409);