Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10587955⋊C3 |
| Hall | #10587955 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ #41 |
| Hall | C5×C7 ⋊ C3×#41 |
| Hall | ΓL(1,𝔽8) ⋊ #709 |
| Sylow | C7 ⋊ C3×#709 |
| Sylow | C5 ⋊ #1159⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C5×C7 | C23×C3 |
| Fitting | C22×C5×C7 | C22×C3 |
| hypercenter | C22 | D5×AGL(1,𝔽7) |
| hypocenter | C5×C7 | C3×#41 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^3,3,2,5,7|a^2=e,g^a=g^6,c^b=ce,f^b=f^4,g^d=g^291 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1159⋊C3, #72218367, AGL(1,𝔽7)×#709 |
| maximal normal subgroup of | D5×#1159⋊C3, #72218367, AGL(1,𝔽7)×#709 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 449);