Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽7) (order 504) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C3×C4×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C4×C7)⋊C32 |
| unambiguous semidirect product | (C3×C4×C7)⋊C3 |
| Fitting | C3×C4×C7 ⋊ C3 |
| derived, Sylow | C7 ⋊ C32×C4 |
| Hall | C4×C7 ⋊ C32 |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3,4)(5,7,6)(8,11,10,9);(1,2,3,5,4,6,7)(12,13,14)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 19);