Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#2091 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C5 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3×C5 |
| socle | C2×C3×C5 | #399 |
| prime radical | C24×C3×C5 | Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C5 ◁ C22×C3×C4×C5 ◁ G | |
444 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(9,10,12,16,13,15,18,22)(11,17,19,23,20,24,21,14);(9,11,18,19,13,20,12,21)(10,14,22,17,15,23,16,24);(10,15)(16,22)390 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 823);