Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C32×C5 ⋊ M16 |
| Sylow | C32 ⋊ #704 |
| Sylow | C5 ⋊ #3711 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×ΓL(1,𝔽16) |
| commutator, Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×C3×C5×D3 | C4 |
| cosocle | C32×Dic5 | C22 |
| hypocenter | C32×C5 | M16 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2^4,3^2,5|a^2=c,c^2=d,b^a=bd,f^a=f^2,g^a=g^3,e^b=e^2,g^c=g^448 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6739, #13088426, #13088428, … |
| maximal normal subgroup of | D3×#6739, #13088426, #13088428, … |
| maximal quotient of | #13085464, (C2×D3)⋊(C3⋊#170), #13085472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 444);