Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×#1359 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | #331 |
| commutator | C42 | C2×C3×C4×C5 |
| Frattini, cosocle | C4×C8 | C22×C3×C5 |
| prime radical | C3×C5×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C3×C4×C5×C8 ◁ G | |
204 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 204 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) = 240 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3)(4,5,6,7,8)(9,10,11,16,19,14,18,22,13,17,21,24,12,15,20,23);(10,14)(11,18)(12,19)(15,17)(16,24)(20,21)345 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 91);