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 |
| direct product | #170×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C3×#170) |
| unambiguous semidirect product | (D7×#170)⋊C3 |
| derived, Hall | C5×C7 ⋊ C2×C3×C8 |
| Hall | D7×#170 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C2×C8 |
| Hall | ΓL(1,𝔽8) ⋊ C2×#170 |
| Sylow | C7 ⋊ C2×C3×#170 |
| 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 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#67195, AGL(1,𝔽7)×D5⋊C8, AGL(1,𝔽7)×#704, … |
| maximal normal subgroup of | AGL(1,𝔽7)×D5⋊C8, AGL(1,𝔽7)×#704, AGL(1,𝔽7)×#700 |
| maximal quotient of | AGL(1,𝔽7)×#67195 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 413);