Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×S5 |
| unambiguous semidirect product | A5⋊(C2×C7) |
| socle | C7×A5 ⋊ C2 |
| derived | A5 ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C7 | S5 |
| commutator, cosocle, hypocenter | A5 | C2×C7 |
| Frattini | C1 | G |
| socle | C7×A5 | C2 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C7 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 35 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4,5,6,7)(8,9,10,11,12);(8,9)49 conjugacy classes — view the full character table.
| maximal subgroup of | D7×S5, C2×C7×S5 |
| maximal normal subgroup of | D7×S5, C2×C7×S5 |
| maximal quotient of | C7×A5⋊C4, C7×#6728, C7×#6729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 135);