Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#1636 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C3×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C5 ◁ C3×C42×C5 ◁ G | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 1920 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3)(4,5,6,7,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19)(25,26,31,35)(27,36,28,39)(29,34,30,33)(32,38,40,37);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,27,31,28)(26,32,35,40)(29,38,30,37)(33,39,34,36);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,28,31,27)(26,32,35,40)(29,37,30,38)(33,39,34,36)390 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 368);