Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C3×C5×#1911 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | #386 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C3×C5 |
| prime radical | C3×C5×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C22×C3×C4×C5 ◁ G | |
724 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)(9,10,13,17,14,16,12,22)(11,18,21,23,20,24,19,15);(9,11)(10,15,16,23)(12,19)(13,21)(14,20)(17,18,22,24);(10,16)(12,13)(15,23)(19,21)435 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 643);