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) = C24 |
| direct product | C3×C5×#2127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | C2×#99 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C2×D4 | C23×C3×C5 |
| prime radical | C3×C5×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C23×C3×C5 ◁ C22×C3×C5×D4 ◁ G | |
572 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)(11,17,18,24)(12,20,22,15)(13,16)(14,19)(21,23);(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(11,18)(14,23)(17,24)(19,21)345 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 859);