Finite Group DatabasebetaFinite Group Databasebeta| order | 7014 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D167 |
| unambiguous semidirect product | C167⋊AGL(1,𝔽7) |
| unambiguous semidirect product | D1169⋊C3 |
| Fitting, socle, derived, Hall | C7×C167 ⋊ C2×C3 |
| Hall | ΓL(1,𝔽8)×C167 ⋊ C2 |
| Hall | D1169 ⋊ C3 |
| Sylow | C167 ⋊ AGL(1,𝔽7) |
| Hall | ΓL(1,𝔽8) ⋊ D167 |
| Sylow | C7 ⋊ C3×D167 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C167 | C2×C3 |
| LCS | G ▷ C7×C167 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,7,167|c^a=c^6,d^a=d^166,c^b=c^2422 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7014, 12);