Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×#1313 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | C42 |
| commutator | C4 | C3×C4×C5×C8 |
| Frattini, cosocle | C4×C8 | C22×C3×C5 |
| socle, prime radical | C22×C3×C5 | #97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ G | |
160 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 1920 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3)(4,5,6,7,8)(17,18,19,23,20,25,27,31,21,24,26,32,28,22,29,30);(9,10,11,13,12,14,15,16)(18,22,24,25)(19,26)(23,31,32,30)(27,29)660 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 45);