Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C2×#11757730 (order 1944) · Inn(G) = C32 · Out(G) = C22×#272 |
| homology | Abelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C32 |
| direct product | C4×C32⋊C9 |
| unambiguous semidirect product | (C32×C4)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C32 |
| commutator | C3 | C3×C4×C9 |
| Frattini, socle, cosocle | C2×C32 | C2×C32 |
| prime radical | C2×C33 | C2×C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ G | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,4,3,5,7,6,8,9)(10,11,13)(12,15,17)(14,18,16)(19,20,21,22);(11,14,15)(13,17,18)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 27);