Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C3×D3×((C3⋊D3)⋊C3)⋊C2 (order 1944) · Inn(G) = H(3,𝔽3) · Out(G) = C22×C3×D3 |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C22×AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C22×C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C32 ◁ G | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(5,6,8)(7,9,12)(10,13,11);(3,4)(6,9,10)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 86);