Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C22×C3×AGL(1,ℤ/9) (order 648) · Inn(G) = D9 · Out(G) = C22×C32 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×D9 |
| unambiguous semidirect product | C9⋊(C22×C7) |
| unambiguous semidirect product | (C2×C9)⋊(C2×C7) |
| Fitting | C2×C7×C9 ⋊ C2 |
| derived, Sylow | C9 ⋊ C22×C7 |
| Hall | C7×C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D9 |
| commutator, cosocle, hypocenter | C9 | C22×C7 |
| Frattini | C3 | C2×D3×C7 |
| Fitting | C2×C7×C9 | C2 |
| socle | C2×C3×C7 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C7 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(3,4,5,6,7,8,9)(11,13)(12,16)(14,17)(15,18);(1,2)(10,15,13,12,17,14,16,11,18)84 conjugacy classes — view the full character table.
| maximal subgroup of | D9×Dic7, #92589, #92591, … |
| maximal normal subgroup of | D9×Dic7, #92589, #92591, … |
| maximal quotient of | C7×Dic18, C4×C7×D9, C7×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 13);