Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×AGL(1,𝔽7))⋊Dic5 |
| unambiguous semidirect product | ((C2×D7)⋊Dic5)⋊C3 |
| Hall | (C2×D7)⋊Dic5 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C22⋊C4 |
| Hall | C5×C7 ⋊ C3×C22⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ C22⋊Dic5 |
| Sylow | C7 ⋊ C3×C22⋊Dic5 |
| Sylow | C5 ⋊ (C2×AGL(1,𝔽7))⋊C4 |
136 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^2,3,2^2,5,7|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.
| maximal subgroup of | D5×(C2×AGL(1,𝔽7))⋊C4, AGL(1,𝔽7)×C22⋊Dic5, #72218154, … |
| maximal normal subgroup of | D5×(C2×AGL(1,𝔽7))⋊C4, AGL(1,𝔽7)×C22⋊Dic5, #72218154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 34);