Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C32×C5 ⋊ D8 |
| Sylow | C32 ⋊ D40 |
| Sylow | C5 ⋊ #3714 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #67151 |
| commutator, Fitting, cosocle | C32×C4×C5 | C22 |
| Frattini, hypercenter | C4 | D3×D15 |
| socle | C2×C32×C5 | D4 |
| hypocenter | C32×C5 | D8 |
| LCS | G ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
104 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^4,3^2,5|b^2=cd,c^2=d,b^a=bc,c^a=cd,f^a=f^2,g^a=g^4,e^b=e^296 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3714, D3×#6653, #13085675, … |
| maximal normal subgroup of | D5×#3714, D3×#6653, #13085675, … |
| maximal quotient of | #13084814, #13084818, #13084819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 268);