Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5×C16 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#1322 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | C22⋊C4 |
| commutator | C22 | C2×C3×C5×C16 |
| Frattini, cosocle | C22×C8 | C22×C3×C5 |
| socle | C22×C3×C5 | C22⋊C8 |
| prime radical | C23×C3×C5 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ C22×C3×C5×C8 ◁ G | |
216 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 960 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3)(4,5,6,7,8)(9,10)(11,15,16,13)(12,14)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(9,11,12,16)(10,13,14,15)660 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 54);