Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | Q8×D5×D7 |
| Fitting | C5×C7×Q8 ⋊ C22 |
| Hall | C5×C7 ⋊ C22×Q8 |
| Sylow | C7 ⋊ C2×Q8×D5 |
| Sylow | C5 ⋊ C2×Q8×D7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D5×D7 |
| commutator, socle, cosocle | C2×C5×C7 | C24 |
| Fitting | C5×C7×Q8 | C22 |
| prime radical | C2×D5×D7 | C22 |
| hypercenter | Q8 | D5×D7 |
| hypocenter | C5×C7 | C22×Q8 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
312 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 280 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | D5×D7×SL(2,𝔽3), Q8×D5×AGL(1,𝔽7), D5×D7⋊SL(2,𝔽3), … |
| maximal normal subgroup of | D5×D7×SL(2,𝔽3), Q8×D5×AGL(1,𝔽7), D5×D7⋊SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 1008);