Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C3×C5×#2135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | C2×#99 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C2×Q8 | C23×C3×C5 |
| prime radical | C2×C3×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×#41 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 480 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3)(4,5,6,7,8)(9,10,14,19,15,20,12,17)(11,21,24,39,25,36,22,38)(13,27,26,37,29,18,30,35)(16,31,32,28,33,40,34,23);(9,11,15,25)(10,16,20,33)(12,24,14,22)(13,28,29,23)(17,34,19,32)(18,36,27,21)(26,31,30,40)(35,38,37,39);(9,12,15,14)(10,17,20,19)(11,22,25,24)(13,26,29,30)(16,32,33,34)(18,35,27,37)(21,38,36,39)(23,31,28,40)345 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 867);