Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C3⋊#157 |
| automorphism | Aut(G) = C2×D3×AΓL(1,𝔽9) (order 1728) · Inn(G) = C3⋊#157 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C3 |
| affine special semilinear | AΣL(1,𝔽81) |
| direct product | C3×C3⋊#157 |
| unambiguous semidirect product | C3⋊(C3×#157) |
| Fitting, socle, Sylow | C34 ⋊ C4 |
| derived | C33 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊#157 |
| commutator, hypocenter | C33 | C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C34 | C4 |
| prime radical | C32×C3⋊D3 | C2 |
| cosocle | C3×C3⋊D3 | C2×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C3 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,11,9,3,2,12,4,6,5,8,7,10);(1,5,2)(3,10,6)(4,9,7)36 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×#157, D3×C3⋊#157, #10620197, … |
| maximal normal subgroup of | C3×D3×#157, D3×C3⋊#157, #10620197, … |
| maximal quotient of | C3×C3⋊#622, C2×C3×C3⋊#157, C3×C9⋊#157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 162);