Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#2170 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C5×C16 | 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×C16 ◁ C2×C3×C5×C16 ◁ G | |
292 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)(9,10,14,18,13,19,28,36,15,20,30,37,29,17,33,35)(11,21,23,26,24,32,38,27,25,34,39,40,22,16,31,12);(9,11)(10,16)(12,18)(13,24)(14,23)(15,25)(17,34)(19,21)(20,32)(22,29)(26,36)(27,37)(28,38)(30,39)(31,33)(35,40);(9,12,14,21,13,26,28,32,15,27,30,34,29,40,33,16)(10,11,18,23,19,24,36,38,20,25,37,39,17,22,35,31)840 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 902);