Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #10603784⋊C3 |
| Hall | #10603784 ⋊ C3 |
| Sylow | C5 ⋊ #1843⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #65688⋊C3 |
| commutator, Fitting, hypocenter | C5×#95 | C3×C4 |
| Frattini | C23 | A4×AGL(1,𝔽5) |
| socle | C23×C5 | C4×A4 |
| prime radical | C5×#95⋊C3 | C4 |
| cosocle | #64937 | C2×C3 |
| LCS | G ▷ C5×#95 | |
| UCS | C1 ◁ C2 | |
122 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2,3,2^3,5,2^3|a^2=c,c^2=gh,d^2=g,e^2=gi,d^a=dh,e^a=ehi,f^a=f^2,d^b=de,e^b=dhi,g^b=ghi,h^b=i,i^b=hi,f^c=f^4,e^d=egh48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237290);