Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C9⋊(C22⋊A4) |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C24×C32 |
| unambiguous semidirect product | C32⋊(C22⋊A4) |
| Fitting | C24×C32 ⋊ C3 |
| Sylow | C24 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C22⋊A4 |
| commutator, socle, cosocle | C24×C3 | C32 |
| Fitting | C24×C32 | C3 |
| hypercenter | C32 | C22⋊A4 |
| hypocenter | C24 | H(3,𝔽3) |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ C32 | |
197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 17 points:
(5,8)(6,7)(10,13,14)(12,16,17);(2,3,4)(6,7,8)(9,10,12)(11,14,16)(13,17,15);(1,4)(2,3)Polycyclic presentation:
3^3,2^4|b^a=bc,d^a=e,e^a=de,f^a=g,g^a=fg56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 555);