Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽7) (order 504) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| Fitting, socle | C2×C3×C7 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C22×C32 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C22 |
| Hall | C2×D7 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2)(7,10,11,8,9,12);(1,2)(3,4,5)(6,7,9,12,11,10,8)42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 28);