Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of AΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C2×C5×AΓL(1,𝔽8) |
| unambiguous semidirect product | AGL(1,𝔽8)⋊(C2×C3×C5) |
| unambiguous semidirect product | (C2×AGL(1,𝔽8))⋊(C3×C5) |
| unambiguous semidirect product | (C2×C5×AGL(1,𝔽8))⋊C3 |
| Fitting, socle, Hall | C24×C5 ⋊ ΓL(1,𝔽8) |
| derived | AGL(1,𝔽8) ⋊ C2×C3×C5 |
| Hall | C2×C5×AGL(1,𝔽8) ⋊ C3 |
| Hall | C2×AGL(1,𝔽8) ⋊ C3×C5 |
| Sylow | C24 ⋊ C5×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | AΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | AGL(1,𝔽8) | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C5 | ΓL(1,𝔽8) |
| LCS | G ▷ AGL(1,𝔽8) | |
| UCS | C1 ◁ C2×C5 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 70 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×C5×AΓL(1,𝔽8), C2×C5×AGL(3,𝔽2), C2×D5×AΓL(1,𝔽8) |
| maximal normal subgroup of | C2×C3×C5×AΓL(1,𝔽8), C2×D5×AΓL(1,𝔽8) |
| maximal quotient of | C2×C3×C5×AΓL(1,𝔽8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 935);