Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #10587852⋊C3 |
| Hall | #10587852 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ Q16 |
| Hall | C5×C7 ⋊ C3×Q16 |
| Hall | ΓL(1,𝔽8) ⋊ Dic20 |
| Sylow | C7 ⋊ C3×Dic20 |
| Sylow | C5 ⋊ #1144⋊C3 |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 560 |
Polycyclic presentation:
2^2,3,2^2,5,7|a^2=d,b^2=e,d^2=e,b^a=bd,g^a=g^6,d^b=de,f^b=f^4,g^c=g^282 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 32);