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×#731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C22×C32 |
| prime radical | C22×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C32 ◁ G | |
70 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)(6,9,15)(7,12,22)(10,19,29)(11,18,26)(13,24,23)(16,20,25)(17,21,28)(27,30,31);(3,4)(5,6,10,8,11,20,14,21,30)(7,9,16,12,18,27,22,28,29)(13,17,19,23,15,25,24,26,31)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 87);