Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #67194 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×#67194 |
| Fitting, socle | C2×C32×C5 ⋊ C8 |
| derived, Hall | C32×C5 ⋊ C2×C8 |
| Sylow | C32 ⋊ C2×#168 |
| Sylow | C5 ⋊ C2×AGL(1,𝔽9) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #67194 |
| commutator, hypocenter | C32×C5 | C2×C8 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C8 |
| prime radical | C2×C5×C3⋊D3 | C4 |
| cosocle | C5×#157 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
54 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
42 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×AGL(1,𝔽9), #13088384, #13088385, … |
| maximal normal subgroup of | Dic5×AGL(1,𝔽9), #13088384, #13088385, … |
| maximal quotient of | #13088622, #13088623, C4×#67194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 801);