Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | C42⋊C27 |
| derived, Sylow | C42 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | C42⋊C3 |
| commutator, hypocenter | C42 | C27 |
| Frattini | C22×C9 | A4 |
| Fitting, cosocle | C42×C9 | C3 |
| socle, prime radical | C22×C3 | C22⋊C9 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C9 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,5)(3,6,10)(4,7,9)(8,11,12)(13,14,17,15,18,23,20,25,30,16,19,24,21,26,31,28,33,36,22,27,32,29,34,37,35,38,39);(1,8,4,3)(5,10,9,12)Polycyclic presentation:
3^3,2^4|a^3=b,b^3=c,d^2=fg,e^2=f,d^a=e,e^a=deg,f^a=g,g^a=fg72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 3);