Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C2×C3×((C3⋊D3)⋊C3)⋊C2 (order 648) · Inn(G) = H(3,𝔽3) · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C4×#731 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | H(3,𝔽3) |
| commutator | C32 | C32×C4 |
| Frattini, cosocle | C2×C32 | C2×C32 |
| socle | C2×C3 | C2×H(3,𝔽3) |
| prime radical | C2×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C32×C4 ◁ G | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4)(5,6,9)(7,13,17)(8,12,18)(10,19,15)(11,22,16)(14,24,27)(20,30,31)(21,29,26)(23,28,25);(5,7,14,8,15,25,16,26,31)(6,10,20,12,21,24,11,13,23)(9,17,27,18,19,28,22,29,30)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 32);