Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C22⋊(C7⋊AGL(1,𝔽5)))⋊C3 |
| Fitting | C22×C5×C7 ⋊ C3×C4 |
| Hall | C22⋊(C7⋊AGL(1,𝔽5)) ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C22⋊C4 |
| Hall | C5×C7 ⋊ C3×C22⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ C22⋊AGL(1,𝔽5) |
| Sylow | C7 ⋊ C3×C22⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ (C22⋊Dic7)⋊C3 |
136 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
Polycyclic presentation:
2^2,3,2^2,5,7|a^2=d,b^a=be,f^a=f^2,g^a=g^6,g^c=g^2,f^d=f^458 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 430);