Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C22×C33 ⋊ C3 |
| Sylow | C22 ⋊ AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center | C3 | C22⋊H(3,𝔽3) |
| commutator, cosocle | C22×C32 | C32 |
| Frattini | C32 | C3×A4 |
| Fitting | C22×C33 | C3 |
| socle | C22×C3 | H(3,𝔽3) |
| hypercenter | C33 | A4 |
| hypocenter | C22 | AΣL(1,𝔽27) |
| LCS | G ▷ C22×C32 ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ C33 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,4)(2,3)(6,9,10);(2,3,4)(5,6,8)(7,9,12)(10,13,11)Polycyclic presentation:
3^4,2^2|b^a=bc,c^a=cd,e^a=f,f^a=ef44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 60);