Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C4 |
| direct product | C3×C5×#1572 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C42 | C23×C3×C5 |
| socle | C22×C3×C5 | C2×C4⋊C4 |
| prime radical | C23×C3×C5 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ C2×C3×C42×C5 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 960 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3)(4,5,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)(27,33)(30,36)(34,39)(37,40)570 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 304);