Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#1953 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C5 | #121 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C3×C5 |
| socle | C2×C3×C5 | #389 |
| prime radical | C23×C3×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C5 ◁ C22×C3×C4×C5 ◁ G | |
368 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)(9,10,14,18,15,19,13,29)(11,20,22,16,24,31,23,32)(12,25,27,33,28,39,26,17)(21,30,35,38,37,40,36,34);(9,11,14,22,15,24,13,23)(10,16,18,31,19,32,29,20)(12,21,27,35,28,37,26,36)(17,30,25,38,33,40,39,34);(9,12,13,26,15,28,14,27)(10,17,18,25,19,33,29,39)(11,21,23,36,24,37,22,35)(16,30,31,38,32,40,20,34)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 685);