Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of C22⋊ΓL(1,𝔽8) |
| automorphism | Aut(G) = C3×A4×AGL(1,𝔽7) (order 1512) · Inn(G) = C22⋊ΓL(1,𝔽8) · Out(G) = C2×C32 |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊(C7⋊C9) |
| unambiguous semidirect product | C7⋊(C22⋊C9) |
| derived, Hall | C22×C7 ⋊ C9 |
| Sylow | C7 ⋊ C22⋊C9 |
| Sylow | C22 ⋊ C7⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C22⋊ΓL(1,𝔽8) |
| commutator, hypocenter | C22×C7 | C9 |
| Fitting, socle, prime radical, cosocle | C22×C3×C7 | C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C3 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3,4)(6,7,8)(9,11,10)(12,13,15,14,16,18,17,19,20);(1,4)(2,3)(5,6,7,9,8,10,11)Polycyclic presentation:
3^2,2^2,7|a^3=b,c^a=d,d^a=cd,e^a=e^236 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 11);