Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C3⋊(C3⋊#157) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C3⋊(C3⋊#157) |
| Fitting, socle, derived, Sylow | C34 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C34 | C4 |
| prime radical, cosocle | C32×C3⋊D3 | C2 |
| LCS | G ▷ C34 | |
| UCS | C1 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,3)(5,6)(7,8)(9,11,12,10);(4,5,6)(7,9,12)(8,11,10);(1,2,3)Polycyclic presentation:
2^2,3^4|a^2=b,c^a=cd^2,d^a=c^2d^2,e^a=e^2,f^a=f^2,c^b=c^2,d^b=d^230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 163);