Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽7) (order 504) · Inn(G) = D7 · Out(G) = C22×C32 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C9×D7 |
| Fitting | C2×C7×C9 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C9 |
| Hall | C7×C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | D7 |
| commutator, hypocenter | C7 | C22×C9 |
| Frattini | C3 | C2×C3×D7 |
| Fitting | C2×C7×C9 | C2 |
| socle | C2×C3×C7 | C2×C3 |
| prime radical | C2×C3×D7 | C3 |
| cosocle | C3×C7 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C9 | |
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)(4,5)(6,7)(8,9)(10,14,18,12,17,13,15,11,16);(1,2)(3,4,6,8,9,7,5)90 conjugacy classes — view the full character table.
| maximal subgroup of | D7×Dic9, #92591, #92592, … |
| maximal normal subgroup of | D7×Dic9, #92591, #92592, … |
| maximal quotient of | C9×Dic14, C4×C9×D7, C9×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 12);