Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C9⋊(C42⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C32×C4 |
| unambiguous semidirect product | C32⋊(C42⋊C3) |
| unambiguous semidirect product | C42⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×C42⋊C3)⋊C3 |
| Fitting | C32×C42 ⋊ C3 |
| Sylow | C42 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C3 | C3×C42⋊C3 |
| commutator, cosocle | C3×C42 | C32 |
| Frattini, socle | C22×C3 | C3×A4 |
| Fitting | C32×C42 | C3 |
| hypercenter | C32 | C42⋊C3 |
| hypocenter | C42 | H(3,𝔽3) |
| LCS | G ▷ C3×C42 ▷ C42 | |
| UCS | C1 ◁ C3 ◁ C32 | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,3,4,8)(5,12,9,10)(14,17,18)(16,20,21);(1,2,5)(3,6,10)(4,7,9)(8,11,12)(13,14,16)(15,18,20)(17,21,19)Polycyclic presentation:
3^3,2^4|d^2=fg,e^2=f,b^a=bc,d^a=e,e^a=deg,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, 103);