Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #10587840⋊C3 |
| Hall | #10587840 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ M16 |
| Hall | C5×C7 ⋊ C3×M16 |
| Hall | ΓL(1,𝔽8) ⋊ #676 |
| Sylow | C7 ⋊ C3×#676 |
| Sylow | C5 ⋊ #1131⋊C3 |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^2,3,2^2,5,7|a^2=d,b^2=d,d^2=e,b^a=be,g^a=g^6,f^b=f^4,g^c=g^294 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1131⋊C3, AGL(1,𝔽7)×#676, #72217890, … |
| maximal normal subgroup of | D5×#1131⋊C3, AGL(1,𝔽7)×#676, #72217890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 20);