Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#4054 |
| Fitting | C5×C7×M16 ⋊ C2 |
| Hall | C5×C7 ⋊ AGL(1,ℤ/8) |
| Sylow | C5 ⋊ C7×AGL(1,ℤ/8) |
136 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4,5,6,7)(8,9,11,12,10)(13,14,15,17,16,18,19,20);(14,18)(17,20);(9,10)(11,12)(14,17)(15,19)(18,20)217 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#4054, C7×D5×AGL(1,ℤ/8), D7×#4054, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#4054, C7×D5×AGL(1,ℤ/8), D7×#4054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 758);