Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C4 |
| direct product | C4×C7×ΓL(1,𝔽16) |
| Fitting | C3×C4×C5×C7 ⋊ C4 |
| derived, Hall | C3×C5 ⋊ C42×C7 |
| Hall | C3×C5×C7 ⋊ C42 |
| Hall | C5×C7 ⋊ C4×Dic3 |
| Hall | C3×C7 ⋊ C4×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C4×C7×Dic3 |
| Sylow | C3 ⋊ C4×C7×AGL(1,𝔽5) |
120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 420 |
252 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D3×C7×AGL(1,𝔽5), C2×C4×C7×ΓL(1,𝔽16) |
| maximal normal subgroup of | C4×D3×C7×AGL(1,𝔽5), C2×C4×C7×ΓL(1,𝔽16) |
| maximal quotient of | C2×C4×C7×ΓL(1,𝔽16) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 639);