Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#1317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C5 | #99 |
| commutator | C23 | C2×C3×C5×C8 |
| Frattini, cosocle | C2×C22⋊C4 | C22×C3×C5 |
| prime radical | C24×C3×C5 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C5 ◁ C23×C3×C5 ◁ C2×C3×C5×C22⋊C4 ◁ G | |
300 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 480 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3)(4,5,6,7,8)(9,10,11,13,12,14,15,16)(17,18)(19,25)(20,28,29,23)(21,24)(22,26,30,32)(27,31);(17,19,21,27)(18,22,24,30)(20,26,29,32)(23,25,28,31)390 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 49);