Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C4×C5×C7 ⋊ C22 |
| Hall | C3×C5×C7 ⋊ C2×D4 |
| Hall | C5×C7 ⋊ D3×D4 |
| Hall | C3×C7 ⋊ D4×D5 |
| Hall | C3×C5 ⋊ D4×D7 |
| Sylow | C7 ⋊ D4×D15 |
| Sylow | C5 ⋊ #66681 |
| Sylow | C3 ⋊ #10587950 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×D15 |
| commutator, socle, cosocle | C2×C3×C5×C7 | C23 |
| Fitting | C3×C4×C5×C7 | C22 |
| hypercenter | C4 | D7×D15 |
| hypocenter | C3×C5×C7 | C2×D4 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
216 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 216 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) = 420 |
Polycyclic presentation:
2^4,3,5,7|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, 718);