Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C5×C7 ⋊ C2×Q8 |
| Hall | C5×C7 ⋊ D3×Q8 |
| Hall | C3×C7 ⋊ Q8×D5 |
| Hall | C3×C5 ⋊ Q8×D7 |
| Sylow | C7 ⋊ Q8×D15 |
| Sylow | C5 ⋊ D21⋊Q8 |
| Sylow | C3 ⋊ D35⋊Q8 |
152 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 840 |
Polycyclic presentation:
2^4,3,5,7|a^2=d,b^2=d,c^2=d,c^a=cd,g^a=g^6,c^b=cd,e^b=e^2,f^b=f^4144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 711);