Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | Aut(G) = C2×C4×((C3⋊D3)⋊C3)⋊C2 (order 864) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C3×C16 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊(C3⋊C16))⋊C3 |
| derived | C32 ⋊ C3×C16 |
| Sylow | H(3,𝔽3) ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C3×C16 |
| Frattini | C3×C8 | C3×D3 |
| Fitting | C8×H(3,𝔽3) | C2 |
| socle | C2×C3 | C3×C3⋊C8 |
| prime radical | C2×H(3,𝔽3) | C8 |
| cosocle | C32×C8 | C2×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C8 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23,25,20,21,24)(19,22);(17,18,20)(19,21,24)(22,25,23)Polycyclic presentation:
2,3,2^3,3^2|a^2=c,c^2=d,d^2=e,f^a=f^2,g^a=g^2,f^b=fg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 30);