Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42×C5 · Schur multiplier (H2) = C4 |
| direct product | C42×C5×ΓL(1,𝔽8) |
| Fitting, Hall | C42×C5×C7 ⋊ C3 |
| derived, Sylow | C7 ⋊ C3×C42×C5 |
| Hall | C42×C7 ⋊ C3×C5 |
| Hall | C5×C7 ⋊ C3×C42 |
120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 560 |
400 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C5×AGL(1,𝔽5)×ΓL(1,𝔽8), C42×C5×AGL(1,𝔽7) |
| maximal normal subgroup of | C42×C5×AGL(1,𝔽7) |
| maximal quotient of | C4×C5×AGL(1,𝔽5)×ΓL(1,𝔽8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 267);