Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D5×AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | #168×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C3×#168) |
| unambiguous semidirect product | (D7×#168)⋊C3 |
| derived, Hall | C5×C7 ⋊ C2×C3×C8 |
| Hall | D7×#168 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C2×C8 |
| Hall | ΓL(1,𝔽8) ⋊ C2×#168 |
| Sylow | C7 ⋊ C2×C3×#168 |
| Sylow | C5 ⋊ C8×AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D5×AGL(1,𝔽7) |
| commutator, hypocenter | C5×C7 | C2×C3×C8 |
| Fitting, cosocle | C4×C5×C7 | C22×C3 |
| socle | C2×C5×C7 | C2×C3×C4 |
| prime radical | C2×C5×AGL(1,𝔽7) | C4 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C4 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 280 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#67194, C8×D5×AGL(1,𝔽7), AGL(1,𝔽7)×#676, … |
| maximal normal subgroup of | C8×D5×AGL(1,𝔽7), AGL(1,𝔽7)×#676, AGL(1,𝔽7)×#681, … |
| maximal quotient of | AGL(1,𝔽7)×#67194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 15);