Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C3⋊(C3⋊(#41⋊C3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊(#41⋊C3)) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×C3⋊D3 |
| commutator, Fitting, cosocle, hypocenter | C32×Q8 | C2×C3 |
| socle | C2×C32 | C2×A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,5,8)(3,13,12,14,9,7,10,15,4,6,11,16)(18,19)(21,22);(1,11,5,9)(2,15,8,13)(3,4,10,12)(6,7,14,16)(20,21,22);(17,18,19)Polycyclic presentation:
2,3,2^2,3^2,2|a^2=g,c^2=g,d^2=g,e^a=e^2,f^a=f^2,c^b=cd,d^b=c,d^c=dg42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 625);