Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C32 |
| direct product | C22×C3×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C32 |
| commutator, Frattini, cosocle | C3 | C22×C33 |
| prime radical | C22×C33 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(5,6,7);(3,4)(8,9,11,10,13,15,14,12,16);(9,12,13)(11,15,16)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 153);