Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×D4×D7 |
| Fitting | C2×C5×C7×D4 ⋊ C2 |
| Hall | C5×C7 ⋊ C22×D4 |
| Sylow | C7 ⋊ C22×C5×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5 | C22×D7 |
| commutator, cosocle | C2×C7 | C24×C5 |
| Frattini | C2 | C23×C5×D7 |
| Fitting | C2×C5×C7×D4 | C2 |
| socle | C22×C5×C7 | C23 |
| hypercenter | C2×C5×D4 | D7 |
| hypocenter | C7 | C22×C5×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×D4 | |
472 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 280 |
250 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C5×D4×AGL(1,𝔽7), C2×D4×D5×D7, C5×D7×AGL(1,ℤ/8), … |
| maximal normal subgroup of | C2×C5×D4×AGL(1,𝔽7), C2×D4×D5×D7, C5×D7×AGL(1,ℤ/8), … |
| maximal quotient of | C5×D7×AGL(1,ℤ/8), C5×D7×#120 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 1031);