Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#1351 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C5 | #99 |
| commutator | C22×C4 | C2×C3×C4×C5 |
| Frattini, cosocle | C2×C22⋊C4 | C22×C3×C5 |
| prime radical | C24×C3×C5 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C5 ◁ C23×C3×C5 ◁ C2×C3×C5×C22⋊C4 ◁ G | |
324 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 480 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3)(4,5,6,7,8)(10,13,14,15)(17,18)(19,23,24,21)(20,22);(9,10)(11,15)(12,14)(13,16)(18,21,22,23)(19,24)345 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 83);