Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of C7⋊S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C2×C7⋊S5 |
| unambiguous semidirect product | (C2×C7)⋊S5 |
| unambiguous semidirect product | (C2×A5)⋊D7 |
| Fitting | C2×C7 ⋊ S5 |
| socle | C2×C7×A5 ⋊ C2 |
| derived | C7×A5 ⋊ C22 |
| Sylow | C7 ⋊ C2×S5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C7⋊S5 |
| commutator, cosocle, hypocenter | C7×A5 | C22 |
| Frattini | C1 | G |
| Fitting | C2×C7 | S5 |
| socle | C2×C7×A5 | C2 |
| LCS | G ▷ C7×A5 | |
| UCS | C1 ◁ C2 | |
114 conjugacy classes of subgroups.
| transitive degree | μt(G) = 70 |
44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 929);