Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C3⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C3⋊(C3⋊C16)) |
| derived, Sylow | C33 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | C3⋊(C3⋊D3) |
| commutator, hypocenter | C33 | C16 |
| Fitting, cosocle | C33×C8 | C2 |
| socle, prime radical | C2×C33 | C8 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C8 | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,19)(21,22)(24,25);(23,24,25);(20,21,22);(17,18,19)Polycyclic presentation:
2^4,3^3|a^2=b,b^2=c,c^2=d,e^a=e^2,f^a=f^2,g^a=g^2120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 231);