Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Dic7⋊ΓL(1,𝔽16) |
| Hall | C3×C5×C7 ⋊ C4⋊C4 |
| Hall | C5×C7 ⋊ C4⋊Dic3 |
| Hall | C3×C7 ⋊ C4⋊AGL(1,𝔽5) |
| Hall | C3×C5 ⋊ Dic7⋊C4 |
| Sylow | C7 ⋊ C4⋊ΓL(1,𝔽16) |
| Sylow | C5 ⋊ Dic7⋊Dic3 |
| Sylow | C3 ⋊ Dic7⋊AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×ΓL(1,𝔽16) |
| commutator, Fitting, socle | C2×C3×C5×C7 | C2×C4 |
| prime radical, cosocle | C2×C3×C7×D5 | C22 |
| hypocenter | C3×C5×C7 | C4⋊C4 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 | |
104 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 420 |
Polycyclic presentation:
2^4,3,5,7|a^2=d,b^2=c,b^a=bd,g^a=g^6,e^b=e^2,f^b=f^2,f^c=f^484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 467);