Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C2×C3×AGL(1,𝔽7) (order 252) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| direct product | C4×C7⋊C9 |
| unambiguous semidirect product | (C4×C7)⋊C9 |
| derived, Sylow | C7 ⋊ C4×C9 |
| Hall | C4×C7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C4×C9 |
| Frattini | C2×C3 | C2×ΓL(1,𝔽8) |
| Fitting | C3×C4×C7 | C3 |
| socle, prime radical, cosocle | C2×C3×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C4 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3,4)(5,7,6)(8,15,14,10,9,16,13,12,11)(17,18,19,20);(1,2,3,5,4,6,7)60 conjugacy classes — view the full character table.
| maximal subgroup of | (C7⋊C8)⋊C9, C8×C7⋊C9, Dic14⋊C9, … |
| maximal normal subgroup of | (C7⋊C8)⋊C9, C8×C7⋊C9, Dic14⋊C9, … |
| maximal quotient of | C8×C7⋊C9, C2×C4×C7⋊C9, C4×C7⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 2);