Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #10587921⋊C3 |
| Hall | #10587921 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ M16 |
| Hall | C5×C7 ⋊ C3×M16 |
| Hall | ΓL(1,𝔽8) ⋊ #704 |
| Sylow | C7 ⋊ C3×#704 |
| 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|b^2=d,d^2=e,b^a=be,g^a=g^6,f^b=f^2,g^c=g^2,f^d=f^452 conjugacy classes — view the full character table.
| maximal subgroup of | #72217787, AGL(1,𝔽7)×#704, #72218301 |
| maximal normal subgroup of | #72217787, AGL(1,𝔽7)×#704, #72218301 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 415);