Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #1113⋊C3 |
| automorphism | Aut(G) = #10620184⋊C2 (order 1296) · Inn(G) = G · Out(G) = C22 |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #1113⋊C3 |
| Fitting, socle | C33 ⋊ A4 |
| derived | #1113 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #1113 | C3 |
| Fitting, socle | C33 | A4 |
| LCS | G ▷ #1113 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 9 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2,4,3,5,7,6,8,9);(3,6)(7,9)Polycyclic presentation:
3,2^2,3^3|b^a=bc,c^a=b,d^a=e,e^a=f,f^a=d,e^b=e^2,f^b=f^2,d^c=d^2,e^c=e^213 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 160);