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×#2171 |
| 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 | C8×D4 |
| prime radical | C3×C5×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C5×C8 ◁ C2×C3×C5×C16 ◁ G | |
288 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)(9,10,14,20,15,19,32,38,16,21,33,23,13,29,31,39)(11,22,25,27,24,17,34,28,18,35,37,26,30,36,40,12);(9,11)(10,17)(12,23)(13,30)(14,25)(15,24)(16,18)(19,35)(20,28)(21,36)(22,29)(26,38)(27,39)(31,40)(32,34)(33,37);(9,12,14,22,15,27,32,17,16,28,33,35,13,26,31,36)(10,18,20,37,19,30,38,40,21,11,23,25,29,24,39,34)660 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 903);