Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×D7)⋊ΓL(1,𝔽16) |
| Hall | C3×C5×C7 ⋊ C22⋊C4 |
| Hall | C5×C7 ⋊ #217 |
| Hall | C3×C7 ⋊ C22⋊AGL(1,𝔽5) |
| Hall | C3×C5 ⋊ (C2×D7)⋊C4 |
| Sylow | C7 ⋊ #6763 |
| Sylow | C5 ⋊ (C2×D7)⋊Dic3 |
| Sylow | C3 ⋊ (C2×D7)⋊AGL(1,𝔽5) |
136 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 420 |
Polycyclic presentation:
2^4,3,5,7|b^2=c,b^a=bd,g^a=g^6,e^b=e^2,f^b=f^2,f^c=f^484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 466);