Finite Group DatabasebetaFinite Group Databasebeta| order | 770 |
| isoclinism | isoclinic class of C11⋊C5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C7 · Schur multiplier (H2) = C1 |
| direct product | C2×C7×C11⋊C5 |
| unambiguous semidirect product | (C2×C11)⋊(C5×C7) |
| unambiguous semidirect product | (C2×C7×C11)⋊C5 |
| Fitting, socle, Hall | C2×C7×C11 ⋊ C5 |
| derived, Sylow | C11 ⋊ C2×C5×C7 |
| Hall | C7×C11 ⋊ C2×C5 |
| Hall | C2×C11 ⋊ C5×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | C11⋊C5 |
| commutator, cosocle, hypocenter | C11 | C2×C5×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C11 | C5 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C2×C7 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 154 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4,5,6,7,8,9)(11,12,14,17,16)(13,15,19,20,18);(10,11,13,16,12,14,18,20,15,17,19)98 conjugacy classes — view the full character table.
| maximal subgroup of | C7×Dic11⋊C5, Dic7×C11⋊C5, C7⋊(Dic11⋊C5), … |
| maximal normal subgroup of | C7×Dic11⋊C5, Dic7×C11⋊C5, C7⋊(Dic11⋊C5), … |
| maximal quotient of | C4×C7×C11⋊C5, C22×C7×C11⋊C5, C2×C3×C7×C11⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(770, 4);