Finite Group DatabasebetaFinite Group Databasebeta| order | 651 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C31 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×C31 |
| unambiguous semidirect product | C7⋊(C3×C31) |
| Fitting, socle, Hall | C7×C31 ⋊ C3 |
| derived, Sylow | C7 ⋊ C3×C31 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C31 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C3×C31 |
| Frattini | C1 | G |
| Fitting, socle | C7×C31 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C31 | |
8 conjugacy classes of subgroups.
| transitive degree | μt(G) = 217 |
155 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×D31, C31×AGL(1,𝔽7), ΓL(1,𝔽8)⋊D31, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×D31, C31×AGL(1,𝔽7), ΓL(1,𝔽8)⋊D31, … |
| maximal quotient of | C2×ΓL(1,𝔽8)×C31, C31×C7⋊C9, C3×ΓL(1,𝔽8)×C31, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(651, 1);