Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Hall | C32×C5 ⋊ C4⋊C4 |
| Sylow | C32 ⋊ C4⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ #3750 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊D3×AGL(1,𝔽5) |
| commutator, Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical, cosocle | C2×C32×D5 | C22 |
| hypocenter | C32×C5 | C4⋊C4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
156 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2^4,3^2,5|a^2=c,b^2=d,b^a=bd,g^a=g^2,e^b=e^2,f^b=f^2,g^c=g^454 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic3⋊AGL(1,𝔽5), #13088408, #13088413, … |
| maximal normal subgroup of | D3×Dic3⋊AGL(1,𝔽5), #13088408, #13088413, … |
| maximal quotient of | #13085423, #13085424, #13085425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 434);