Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of D3×AGL(1,𝔽7) |
| automorphism | Aut(G) = D3×AGL(1,𝔽7) (order 252) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| Fitting, socle, derived | C3×C7 ⋊ C22×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C22 |
| Sylow | C7 ⋊ C2×C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7 | C22×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 21 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2,3)(5,8,9,6,7,10);(2,3)(4,5,7,10,9,8,6)21 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 26);