Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of AGL(1,𝔽5)×AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #10587920⋊C3 |
| derived, Hall | C5×C7 ⋊ C2×C3×C8 |
| Hall | #10587920 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C2×C8 |
| Hall | ΓL(1,𝔽8) ⋊ D5⋊C8 |
| Sylow | C7 ⋊ C3×D5⋊C8 |
| Sylow | C5 ⋊ C8×AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽5)×AGL(1,𝔽7) |
| commutator, hypocenter | C5×C7 | C2×C3×C8 |
| Fitting, socle | C2×C5×C7 | C2×C3×C4 |
| prime radical | C2×C5⋊AGL(1,𝔽7) | C4 |
| cosocle | C7×Dic5 | C22×C3 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C2 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^2,3,2^2,5,7|a^2=e,b^2=d,d^2=e,g^a=g^6,f^b=f^2,g^c=g^2,f^d=f^470 conjugacy classes — view the full character table.
| maximal subgroup of | #72217456, AGL(1,𝔽7)×D5⋊C8, #72218296, … |
| maximal normal subgroup of | AGL(1,𝔽7)×D5⋊C8, #72218296, #72218301 |
| maximal quotient of | #72217456 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 414);