Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C3×C5×#1913 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | #386 |
| commutator | C23 | C22×C3×C4×C5 |
| Frattini, cosocle | C24 | C23×C3×C5 |
| prime radical | C3×C5×#534 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C24×C3×C5 ◁ G | |
760 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)(11,14,18,22)(12,19,20,15)(13,16)(17,21)(23,24);(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,24)(20,23);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24)435 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 645);