Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×D5×#157 |
| Fitting, socle | C2×C32×C5 ⋊ C2×C4 |
| derived, Hall | C32×C5 ⋊ C22×C4 |
| Sylow | C32 ⋊ C2×C4×D5 |
| Sylow | C5 ⋊ C22×#157 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×#157 |
| commutator, hypocenter | C32×C5 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×D5×C3⋊D3 | C2 |
| cosocle | C5×C3⋊D3 | C23 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
172 conjugacy classes of subgroups.
| transitive degree | μt(G) = 60 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×AGL(1,𝔽5))⋊#157, #13088362, #13088369, … |
| maximal normal subgroup of | (C2×AGL(1,𝔽5))⋊#157, #13088362, #13088369, … |
| maximal quotient of | #157×Dic10, #13088773, #13088774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 807);