Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#2153 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C5×C8 | C2×D4 |
| commutator | C4 | C22×C3×C5×C8 |
| Frattini, cosocle | C2×C8 | C23×C3×C5 |
| socle | C2×C3×C5 | C2×C4⋊C8 |
| prime radical | C23×C3×C5 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C5×C8 ◁ C22×C3×C5×C8 ◁ G | |
304 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)(9,10,13,18,14,17,27,32,15,19,28,33,12,25,26,38)(11,20,23,35,24,30,37,39,21,34,36,40,22,16,29,31);(9,11,12,22,15,21,14,24)(10,16,17,20,19,30,25,34)(13,23,26,29,28,36,27,37)(18,31,32,35,33,39,38,40);(11,21)(16,30)(20,34)(22,24)(23,36)(29,37)(31,39)(35,40)660 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 885);