Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×C5×#2126 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C5 | #99 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C2×D4 | C23×C3×C5 |
| socle | C2×C3×C5 | C2×#99 |
| prime radical | C3×C5×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×#41 ◁ G | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3)(4,5,6,7,8)(9,10)(11,15)(12,16,20,14)(13,17)(18,22,24,21)(19,23);(10,14)(12,20)(15,21)(16,17)(18,24)(22,23);(9,11,13,19)(10,15,17,23)(12,18,20,24)(14,21,16,22)390 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 858);