Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C3×D3×AGL(1,𝔽7) (order 756) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×C3×D3 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C22×C7⋊C9 |
| derived, Sylow | C7 ⋊ C22×C9 |
| Hall | C22×C7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C22×C9 |
| Frattini | C3 | C22×ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C22×C3×C7 | C3 |
| cosocle | C3×C7 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C3 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 2 generators on 20 points:
(3,4)(6,7,8)(9,11,10)(12,16,20,14,19,15,17,13,18);(1,2)(5,6,7,9,8,10,11)60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×Dic7⋊C9, (C22⋊D7)⋊C9, C2×C4×C7⋊C9, … |
| maximal normal subgroup of | C2×Dic7⋊C9, (C22⋊D7)⋊C9, C2×C4×C7⋊C9, … |
| maximal quotient of | C2×C4×C7⋊C9, D4×C7⋊C9, Q8×C7⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 9);