Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #10608437⋊C3 |
| Hall | #10608437 ⋊ C3 |
| Sylow | C5 ⋊ #2657⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×#127⋊C3 |
| commutator, cosocle, hypocenter | C5×#95 | C22×C3 |
| Frattini | C23 | C2×D5×A4 |
| Fitting | C5×#392 | C2×C3 |
| socle | C23×C5 | C22×A4 |
| LCS | G ▷ C5×#95 | |
| UCS | C1 ◁ C2 | |
338 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^2,3,2^2,5,2^3|a^2=gh,b^2=gh,d^2=g,e^2=gi,b^a=bgh,f^a=f^4,d^b=dh,e^b=ehi,d^c=de,e^c=dhi,g^c=ghi,h^c=i,i^c=hi,e^d=egh60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237819);