Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4174 |
| automorphism | Aut(G) = C2×C3×((C3⋊D3)⋊C3)⋊C2 (order 648) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×#4174 |
| Fitting | C2×AΣL(1,𝔽27) ⋊ C2 |
| derived | H(3,𝔽3) ⋊ C22×C3 |
| Sylow | AΣL(1,𝔽27) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C3 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C22×C3 |
| Frattini | C32 | C2×C3×D3 |
| Fitting | C2×AΣL(1,𝔽27) | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 11 points:
(3,4,6)(5,11,7,8,9,10);(1,2)(3,5,8)(4,7,10)(6,9,11)44 conjugacy classes — view the full character table.
| maximal subgroup of | #4174⋊C4, #10619549, #10619550, … |
| maximal normal subgroup of | #4174⋊C4, #10619549, #10619550, … |
| maximal quotient of | #10619607, C4×#4174, C4⋊#4174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 68);