Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D5×#611 |
| Fitting | C22×C5×C9 ⋊ C22 |
| Hall | C5×C9 ⋊ C2×D4 |
| Sylow | C9 ⋊ D4×D5 |
| Sylow | C5 ⋊ C2×#611 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D5×D9 |
| commutator, cosocle | C2×C5×C9 | C23 |
| Frattini | C2×C3 | C2×D3×D5 |
| Fitting | C22×C5×C9 | C22 |
| socle | C2×C3×C5 | C22×D3 |
| hypercenter | C22 | D5×D9 |
| hypocenter | C5×C9 | C2×D4 |
| LCS | G ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
162 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#611, #611⋊AGL(1,𝔽5), D5×#3697, … |
| maximal normal subgroup of | AGL(1,𝔽5)×#611, #611⋊AGL(1,𝔽5), D5×#3697, … |
| maximal quotient of | #13084137, #13084138, #13084154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 149);