Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #10624311 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, Hall | C32×C5 ⋊ SD16 |
| Sylow | C32 ⋊ #687 |
| Sylow | C5 ⋊ AΓL(1,𝔽9) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C5×#157 | C22 |
| Fitting, socle, hypocenter | C32×C5 | SD16 |
| prime radical | C5×GO+(4,𝔽2) | C2 |
| LCS | G ▷ C5×#157 ▷ C5×C3⋊D3 ▷ C32×C5 | |
| UCS | C1 | |
50 conjugacy classes of subgroups.
| transitive degree | μt(G) = 45 |
Polycyclic presentation:
2^4,3^2,5|a^2=c,c^2=d,b^a=bcd,e^a=ef,f^a=e,g^a=g^4,c^b=cd,e^b=ef^2,f^b=f^2,e^c=e^2f,f^c=ef,e^d=e^2,f^d=f^227 conjugacy classes — view the full character table.
| maximal subgroup of | D5×AΓL(1,𝔽9), C2×#10624311, C3×#10624311 |
| maximal normal subgroup of | D5×AΓL(1,𝔽9), C2×#10624311, C3×#10624311 |
| maximal quotient of | GO+(4,𝔽2)⋊Dic5, #13088394, #13088395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 780);