Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4183 |
| automorphism | Aut(G) = C2×(#4174⋊C2)⋊C3 (order 1944) · Inn(G) = #4183 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | C2×#4183 |
| Fitting | C2×AΣL(1,𝔽27) ⋊ C2 |
| derived, Sylow | AΣL(1,𝔽27) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4183 |
| commutator, cosocle, hypocenter | AΣL(1,𝔽27) | C22 |
| Frattini | C32 | C2×C3⋊D3 |
| Fitting | C2×AΣL(1,𝔽27) | C2 |
| socle | C2×C3 | #272 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C2 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,2)(4,7)(5,6)(8,11)(9,10);(1,2)(3,5,6);(3,4,7)(5,8,10)(6,9,11)26 conjugacy classes — view the full character table.
| maximal subgroup of | #4183⋊C4, #10619549, #10619551, … |
| maximal normal subgroup of | #4183⋊C4, #10619549, #10619551, … |
| maximal quotient of | #10619660, C4×#4183, C4⋊#4183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 77);