Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C3×C5×#2134 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | C2×#99 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C2×Q8 | C23×C3×C5 |
| prime radical | C3×C5×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×#41 ◁ G | |
524 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)(9,10,13,18,14,19,11,16)(12,15,20,23,21,17,22,24);(10,15)(11,13)(12,21)(16,24)(17,19)(18,23);(9,11,14,13)(10,16,19,18)(12,20,21,22)(15,23,17,24)345 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 866);