Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C9⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C9⋊#170 |
| unambiguous semidirect product | (C2×C9)⋊#170 |
| derived, Hall | C5×C9 ⋊ C2×C8 |
| Sylow | C9 ⋊ C2×#170 |
| Sylow | C5 ⋊ C2×C9⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C9⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×C9 | C2×C8 |
| Frattini | C2×C3 | C2×ΓL(1,𝔽16) |
| Fitting | C22×C5×C9 | C4 |
| socle, prime radical | C22×C3×C5 | Dic3 |
| cosocle | C9×Dic5 | C22 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C22 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | Dic9×#170, Dic9⋊#170, #13083930, … |
| maximal normal subgroup of | Dic9×#170, Dic9⋊#170, #13083930, … |
| maximal quotient of | C2×C9⋊#674, #13083986, C4×C9⋊#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 122);