Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C2×C32 |
| direct product | C22×C32⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C32 |
| commutator | C3 | C22×C3×C9 |
| Frattini, cosocle | C32 | C22×C32 |
| prime radical | C22×C33 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(6,8,11)(9,13,12)(14,15,17,16,18,20,19,21,22);(3,4)(5,6,9)(7,8,12)(10,11,13)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 82);