Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C9⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C2×C32 |
| unambiguous semidirect product | (C3×A4)⋊C9 |
| Fitting | C22×C3×C9 ⋊ C3 |
| Sylow | C22 ⋊ C32⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C3×A4 |
| commutator | C22×C3 | C3×C9 |
| Fitting | C22×C3×C9 | C3 |
| socle, cosocle | C22×C32 | C32 |
| prime radical | C32×A4 | C3 |
| hypercenter | C3×C9 | A4 |
| hypocenter | C22 | C32⋊C9 |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C32 ◁ C3×C9 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,4)(2,3)(5,6,8,7,9,11,10,12,13)(15,18,19)(17,21,22);(2,3,4)(14,15,17)(16,19,21)(18,22,20)Polycyclic presentation:
3^4,2^2|b^3=c,b^a=bd,e^a=f,f^a=ef60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 52);