Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×#1405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C3×C4×C5 |
| Frattini, cosocle | C4⋊Q8 | C22×C3×C5 |
| prime radical | C3×C5×2−1+4 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C22×C3×C5 ◁ C2×C3×C4×C5 ◁ C3×C5×C4⋊Q8 ◁ G | |
212 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 2 generators on 24 points:
(1,2,3)(4,5,6,7,8)(9,10,12,16,14,11,18,20)(13,17,23,15,24,21,22,19);(9,11)(10,14)(12,15)(13,17)(16,23)(18,19)(20,22)(21,24)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 137);