Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of AGL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C24⋊C5)⋊C9 |
| unambiguous semidirect product | (C22⋊(C22⋊C9))⋊C5 |
| derived, Sylow | C24 ⋊ C5×C9 |
| Hall | C22⋊(C22⋊C9) ⋊ C5 |
| Hall | C24⋊C5 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | AGL(1,𝔽16) |
| commutator, hypocenter | C24 | C5×C9 |
| Fitting, socle, cosocle | C24×C3 | C3×C5 |
| prime radical | C3×C24⋊C5 | C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C3 | |
23 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
Polycyclic presentation:
3,5,3,2^4|a^3=c,d^a=e,e^a=de,f^a=dfg,g^a=ef,d^b=defg,e^b=eg,f^b=ef,g^b=def48 conjugacy classes — view the full character table.
| maximal subgroup of | ((C24⋊C5)⋊C2)⋊C9, C2×(C24⋊C5)⋊C9 |
| maximal normal subgroup of | ((C24⋊C5)⋊C2)⋊C9, C2×(C24⋊C5)⋊C9 |
| maximal quotient of | C2×(C24⋊C5)⋊C9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 391);