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×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C32 |
| commutator, Frattini, cosocle | C3 | C22×C33 |
| socle | C22×C3 | C33 |
| prime radical | C22×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(6,9,15)(10,18,16)(11,17,23)(19,26,24)(20,25,29)(27,31,30);(3,4)(5,6,10)(7,11,19)(8,9,16)(12,20,27)(13,17,24)(14,15,18)(21,25,30)(22,23,26)(28,29,31);(5,7,12,8,13,21,14,22,28)(6,11,20,9,17,25,15,23,29)(10,19,27,16,24,30,18,26,31)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 154);