Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (ΓL(1,𝔽8)⋊C4)⋊Dic5 |
| unambiguous semidirect product | (Dic7⋊Dic5)⋊C3 |
| Hall | Dic7⋊Dic5 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C4⋊C4 |
| Hall | C5×C7 ⋊ C3×C4⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ C4⋊Dic5 |
| Sylow | C7 ⋊ C3×C4⋊Dic5 |
| Sylow | C5 ⋊ (ΓL(1,𝔽8)⋊C4)⋊C4 |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 560 |
Polycyclic presentation:
2^2,3,2^2,5,7|a^2=e,b^2=d,b^a=be,g^a=g^6,f^b=f^4,g^c=g^294 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 37);