Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C5×C9 ⋊ #41 |
| Sylow | C9 ⋊ #709 |
| Sylow | C5 ⋊ #3697 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D5×D9 |
| commutator, cosocle | C2×C5×C9 | C23 |
| Frattini | C2×C3 | C2×D3×D5 |
| Fitting | C4×C5×C9 | C22 |
| socle | C2×C3×C5 | C22×D3 |
| hypocenter | C5×C9 | #41 |
| LCS | G ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C4 | |
120 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^4,3,5,3|c^2=d,e^3=g,b^a=bd,f^a=f^4,e^b=e^2g^2,g^b=g^290 conjugacy classes — view the full character table.
| maximal subgroup of | #13084013, #13084014, #13084015, … |
| maximal normal subgroup of | #13084013, #13084014, #13084015, … |
| maximal quotient of | #13084083, #13084084, #13084089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 132);