Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of A42 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | (C22⋊A4)⋊C9 |
| derived, Sylow | C24 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | A42 |
| commutator, hypocenter | C24 | C3×C9 |
| Fitting, socle, cosocle | C24×C3 | C32 |
| prime radical | C3×C22⋊A4 | C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C3 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,4,3)(5,8)(6,7)(9,11,15,12,16,13,17,10,14);(1,4)(2,3)(6,8,7)(9,10,13,12,11,14,17,16,15)Polycyclic presentation:
3^3,2^4|a^3=c,b^3=c,f^a=fg,g^a=f,d^b=de,e^b=d48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 525);