Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C7 · Schur multiplier (H2) = C22 |
| direct product | C5×C7×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5×C7 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C23×C5×C7 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C7 ◁ C2×C4×C5×C7 ◁ G | |
136 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 80 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)(6,7,8,9,10,11,12)(13,14)(15,17)(16,18)(19,20);(14,17)(15,20)(18,19);(14,18)(17,19)385 conjugacy classes — view the full character table.
| maximal subgroup of | C5×ΓL(1,𝔽8)×AGL(1,ℤ/8), C7×D5×AGL(1,ℤ/8), C5×D7×AGL(1,ℤ/8), … |
| maximal normal subgroup of | C5×ΓL(1,𝔽8)×AGL(1,ℤ/8), C7×D5×AGL(1,ℤ/8), C5×D7×AGL(1,ℤ/8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 912);