Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3×ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | D3×C3⋊#170 |
| unambiguous semidirect product | (C3×D3)⋊#170 |
| derived, Hall | C32×C5 ⋊ C2×C8 |
| Sylow | C32 ⋊ C2×#170 |
| Sylow | C5 ⋊ D3×C3⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×ΓL(1,𝔽16) |
| commutator, hypocenter | C32×C5 | C2×C8 |
| Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×C3×C5×D3 | C4 |
| cosocle | C32×Dic5 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
96 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | D32×#170, D3×#6738, D3×#6739, … |
| maximal normal subgroup of | D32×#170, D3×#6738, D3×#6739, … |
| maximal quotient of | D3×C3⋊#674, #13085460, Dic3×C3⋊#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 442);