Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10587856⋊C3 |
| Hall | #10587856 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C22⋊C4 |
| Hall | C5×C7 ⋊ C3×C22⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ #685 |
| Sylow | C7 ⋊ C3×#685 |
| Sylow | C5 ⋊ (C2×AGL(1,𝔽7))⋊C4 |
136 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 45 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|a^2=e,b^2=de,b^a=bd,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, #72218149, AGL(1,𝔽7)×#685, … |
| maximal normal subgroup of | D5×(C2×AGL(1,𝔽7))⋊C4, #72218149, AGL(1,𝔽7)×#685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 36);