Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of (#126⋊C3)⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (#126⋊C3)⋊AGL(1,𝔽5) |
| unambiguous semidirect product | #10617360⋊C3 |
| Fitting | C5×#126 ⋊ C3×C4 |
| derived | C42×C5 ⋊ C2×C3×C4 |
| Hall | #10617360 ⋊ C3 |
| Sylow | C5 ⋊ (#126⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C42×C5 | C2×C3×C4 |
| Frattini | C22 | C2×A4×AGL(1,𝔽5) |
| Fitting | C5×#126 | C3×C4 |
| socle | C22×C5 | C2×C4×A4 |
| prime radical | D5×#126⋊C3 | C2 |
| cosocle | C42×D5 | C22×C3 |
| LCS | G ▷ C42×C5 | |
| UCS | C1 | |
232 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^2,3,2^3,5,2^2|a^2=d,e^2=h,f^2=hi,e^a=eh,f^a=fhi,g^a=g^2,e^b=ehi,f^b=fi,e^c=ef,f^c=ehi,h^c=i,i^c=hi,g^d=g^444 conjugacy classes — view the full character table.
| maximal subgroup of | #72242815, AGL(1,𝔽5)×#560⋊C3 |
| maximal normal subgroup of | AGL(1,𝔽5)×#560⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 238233);