Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of C3⋊AGL(1,𝔽7) |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽7) (order 504) · Inn(G) = C3⋊AGL(1,𝔽7) · Out(G) = C22 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(ΓL(1,𝔽8)⋊C4) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊Dic3 |
| unambiguous semidirect product | Dic21⋊C3 |
| derived | C3×C7 ⋊ C3×C4 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C4 |
| Sylow | C7 ⋊ C3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊AGL(1,𝔽7) |
| commutator, hypocenter | C3×C7 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C7 | C2×C3 |
| prime radical | C2×C3×ΓL(1,𝔽8) | C2 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,5,6,3,4,7)(9,10)(11,12,13,14);(1,2,4,7,6,5,3)(8,9,10)Polycyclic presentation:
2,3,2,3,7|a^2=c,d^a=d^2,e^a=e^6,e^b=e^224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 18);