Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of C3⋊(C7⋊AGL(1,𝔽5)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D5⋊(C3⋊(C7⋊C8)) |
| derived, Hall | C3×C5×C7 ⋊ C2×C8 |
| Hall | C5×C7 ⋊ C2×C3⋊C8 |
| Hall | C3×C7 ⋊ D5⋊C8 |
| Hall | C3×C5 ⋊ C2×C7⋊C8 |
| Sylow | C7 ⋊ D5⋊(C3⋊C8) |
| Sylow | C5 ⋊ C2×C3⋊(C7⋊C8) |
| Sylow | C3 ⋊ D5⋊(C7⋊C8) |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 840 |
Polycyclic presentation:
2^4,3,5,7|a^2=c,b^2=d,c^2=d,e^a=e^2,f^a=f^2,g^a=g^6,f^c=f^4180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 644);