Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×C5×#1812 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C5 | #117 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C3×C5 |
| socle | C2×C3×C5 | #380 |
| prime radical | C2×C3×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C5 ◁ C2×C3×C5×M16 ◁ G | |
520 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)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,27)(20,33,35,31)(22,32)(24,30,39,38)(28,34,37,40)(29,36);(9,11,13,21,14,22,27,36)(10,15,17,19,18,29,26,32)(12,20,24,34,25,35,39,40)(16,28,30,31,23,37,38,33);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,31)(21,34)(22,35)(26,38)(27,39)(29,37)(32,33)(36,40)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 544);