Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4175 |
| automorphism | Aut(G) = C2×(#4174⋊C2)⋊C3 (order 1944) · Inn(G) = #4175 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×#4175 |
| Fitting | C2×AΣL(1,𝔽27) ⋊ C2 |
| derived | C33 ⋊ C22×C3 |
| Sylow | AΣL(1,𝔽27) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4175 |
| commutator, cosocle, hypocenter | C33 | C22×C3 |
| Frattini | C32 | C2×C3×D3 |
| Fitting | C2×AΣL(1,𝔽27) | C2 |
| socle | C2×C3 | (C3⋊D3)⋊C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2 | |
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)(5,8,10,9,7,11);(1,2)(4,7,8)26 conjugacy classes — view the full character table.
| maximal subgroup of | #4175⋊C4, #10619550, #10619551, … |
| maximal normal subgroup of | #4175⋊C4, #10619550, #10619551, … |
| maximal quotient of | #10619608, C4×#4175, C4⋊#4175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 69);