Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = C2×C3×AGL(1,𝔽7) (order 252) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×D7⋊C9 |
| unambiguous semidirect product | D7⋊(C2×C9) |
| unambiguous semidirect product | (C2×D7)⋊C9 |
| unambiguous semidirect product | (C7⋊C9)⋊C22 |
| unambiguous semidirect product | (C2×C7⋊C9)⋊C2 |
| derived, Sylow | C7 ⋊ C22×C9 |
| Hall | C7⋊C9 ⋊ C22 |
| Hall | C2×D7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C22×C9 |
| Frattini | C3 | C2×AGL(1,𝔽7) |
| Fitting, socle | C2×C3×C7 | C2×C3 |
| prime radical | C2×C3×D7 | C3 |
| cosocle | C3×C7 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3 | |
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:
(4,8,7,5,6,9)(10,14,18,12,17,13,15,11,16);(1,2)(3,4,7,8,6,9,5)42 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D7⋊C9, C4⋊(D7⋊C9), (C22⋊D7)⋊C9, … |
| maximal normal subgroup of | C4×D7⋊C9, C4⋊(D7⋊C9), (C22⋊D7)⋊C9, … |
| maximal quotient of | Dic14⋊C9, C4×D7⋊C9, C4⋊(D7⋊C9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 7);