Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C22×#733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C22×C32 |
| prime radical | C22×C32 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C32 ◁ G | |
65 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)(5,6,9,8,12,18,16,11,22)(7,13,17,15,10,19,26,21,29)(14,24,27,25,23,28,31,20,30);(3,4)(5,7,14,16,26,31,8,15,25)(6,10,20,11,13,23,12,21,24)(9,17,27,22,29,30,18,19,28)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 89);