Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C4×C5×C7 ⋊ C22 |
| Hall | C5×C7 ⋊ #120 |
| Sylow | C7 ⋊ #4028 |
| Sylow | C5 ⋊ #6500 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D5×D7 |
| commutator, socle, cosocle | C22×C5×C7 | C23 |
| Fitting | C2×C4×C5×C7 | C22 |
| hypercenter | C2×C4 | D5×D7 |
| hypocenter | C5×C7 | #120 |
| LCS | G ▷ C22×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
260 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^5,5,7|c^2=d,c^a=ce,g^a=g^6,c^b=cd,f^b=f^4112 conjugacy classes — view the full character table.
| maximal subgroup of | #72218360, D5×#6500, D7×#4028, … |
| maximal normal subgroup of | #72218360, D5×#6500, D7×#4028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 529);