Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = D3×#4208 (order 972) · Inn(G) = C32 · Out(G) = C3×D32 |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C2 |
| direct product | C22×C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C32 |
| commutator | C3 | C22×C3×C9 |
| Frattini, cosocle | C9 | C22×C32 |
| socle | C22×C3 | C3×C9 |
| prime radical | C22×C32 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2)(5,6,9,7,11,17,13,20,25,8,12,16,14,21,24,22,28,30,15,10,18,23,19,26,29,27,31);(3,4)(6,10,12)(9,16,18)(11,19,21)(17,24,26)(20,27,28)(25,30,31)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 85);