Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of D27⋊C3 |
| automorphism | Aut(G) = C2×AGL(1,ℤ/27) (order 972) · Inn(G) = D27⋊C3 · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×D27⋊C3 |
| unambiguous semidirect product | D27⋊(C2×C3) |
| unambiguous semidirect product | (C27⋊C3)⋊C22 |
| unambiguous semidirect product | (C2×D27)⋊C3 |
| unambiguous semidirect product | (C2×C27⋊C3)⋊C2 |
| Fitting | C2×C27⋊C3 ⋊ C2 |
| derived | C27 ⋊ C22×C3 |
| Sylow | C27⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D27⋊C3 |
| commutator, cosocle, hypocenter | C27 | C22×C3 |
| Frattini | C9 | C2×C3×D3 |
| Fitting | C2×C27⋊C3 | C2 |
| socle | C2×C3 | C3×D9 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C2 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 29 points:
(4,15,18,7,8,17)(5,11)(6,14)(9,20,19,10,21,28)(12,13)(16,29,27,23,26,22)(24,25);(1,2)(3,23,17,24,28,21,5,4,27,6,29,15,12,10,9,13,8,26,14,22,7,11,20,19,25,18,16)42 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D27⋊C3, C4⋊(D27⋊C3), #6171⋊C3, … |
| maximal normal subgroup of | C4×D27⋊C3, C4⋊(D27⋊C3), #6171⋊C3, … |
| maximal quotient of | (C27⋊C3)⋊Q8, C4×D27⋊C3, C4⋊(D27⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 67);