Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = C2×C3×AGL(1,𝔽7) (order 252) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Dic7⋊C9 |
| unambiguous semidirect product | (C7⋊C9)⋊C4 |
| derived, Sylow | C7 ⋊ C4×C9 |
| Hall | C7⋊C9 ⋊ C4 |
| Hall | Dic7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C4×C9 |
| Fitting, socle, prime radical, cosocle | C2×C3×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3 | |
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:
(2,6,5,3,4,7)(8,15,14,10,9,16,13,12,11)(17,18,19,20);(1,2,5,6,4,7,3)Polycyclic presentation:
2,3,2,3,7|a^2=c,b^3=d,e^a=e^6,e^b=e^442 conjugacy classes — view the full character table.
| maximal subgroup of | Dic14⋊C9, C4×D7⋊C9, C2×Dic7⋊C9, … |
| maximal normal subgroup of | Dic14⋊C9, C4×D7⋊C9, C2×Dic7⋊C9, … |
| maximal quotient of | (C7⋊C8)⋊C9, C2×Dic7⋊C9, Dic7⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 1);