Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #3714 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#3714 |
| Hall | C32×C5 ⋊ D8 |
| Sylow | C32 ⋊ C5×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | #626 |
| commutator, cosocle | C32×C4 | C22×C5 |
| Frattini | C4 | C5×D32 |
| Fitting | C32×C4×C5 | C22 |
| socle | C2×C32×C5 | D4 |
| hypercenter | C4×C5 | D32 |
| hypocenter | C32 | C5×D8 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 | |
104 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3714, #13085672, #13085674, … |
| maximal normal subgroup of | D5×#3714, #13085672, #13085674, … |
| maximal quotient of | C5×#63546, C5×#63548, C5×#63549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 250);