Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C128 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#10597921 |
| derived, Hall | C3×C5 ⋊ C128 |
| Sylow | C5 ⋊ C3⋊C128 |
| Sylow | C3 ⋊ #10597921 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C128 |
| Fitting | C3×C5×C32 | C4 |
| socle, prime radical | C2×C3×C5 | C64 |
| cosocle | C3×#64663 | C2 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C32 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 136 |
| transitive degree | μt(G) = 1920 |
Polycyclic presentation:
2^7,3,5|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^a=h^2,i^a=i^2,i^b=i^4288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 2333);