Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D5×AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
| direct product | Dic5×ΓL(1,𝔽8)⋊C4 |
| unambiguous semidirect product | (Dic5×Dic7)⋊C3 |
| derived, Hall | C5×C7 ⋊ C3×C42 |
| Hall | Dic5×Dic7 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C42 |
| Hall | ΓL(1,𝔽8) ⋊ C4×Dic5 |
| Sylow | C7 ⋊ C3×C4×Dic5 |
| Sylow | C5 ⋊ C4×ΓL(1,𝔽8)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D5×AGL(1,𝔽7) |
| commutator, hypocenter | C5×C7 | C3×C42 |
| Fitting, socle, cosocle | C22×C5×C7 | C22×C3 |
| prime radical | C22×C5×ΓL(1,𝔽8) | C22 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C22 | |
120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 560 |
112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 33);