Finite Group DatabasebetaFinite Group Databasebeta| order | 735 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C7 · Schur multiplier (H2) = C1 |
| direct product | C5×C7×ΓL(1,𝔽8) |
| unambiguous semidirect product | C7⋊(C3×C5×C7) |
| unambiguous semidirect product | (C5×C7)⋊(C3×C7) |
| Fitting, socle, Hall | C5×C72 ⋊ C3 |
| derived | C7 ⋊ C3×C5×C7 |
| Sylow | C72 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C7 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C3×C5×C7 |
| Frattini | C1 | G |
| Fitting, socle | C5×C72 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C5×C7 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 105 |
175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(735, 3);