Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×M16×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C7×M16)⋊(C3×C5) |
| unambiguous semidirect product | (C5×C7×M16)⋊C3 |
| Fitting, Hall | C5×C7×M16 ⋊ C3 |
| Hall | C7×M16 ⋊ C3×C5 |
| Hall | C5×C7 ⋊ C3×M16 |
| Sylow | C7 ⋊ C3×C5×M16 |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 280 |
250 conjugacy classes — view the full character table.
| maximal subgroup of | D5×M16×ΓL(1,𝔽8), C5×M16×AGL(1,𝔽7), M16×C5⋊AGL(1,𝔽7), … |
| maximal normal subgroup of | D5×M16×ΓL(1,𝔽8), C5×M16×AGL(1,𝔽7), M16×C5⋊AGL(1,𝔽7), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 271);