Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×#1332 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | D8 |
| commutator | C8 | C2×C3×C5×C8 |
| Frattini, cosocle | C4×C8 | C22×C3×C5 |
| socle, prime radical | C22×C3×C5 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ C3×C42×C5 ◁ C3×C4×C5×C8 ◁ G | |
196 conjugacy classes of subgroups.
| 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)(9,10,11,13,12,14,15,16)(18,22)(19,25)(20,27)(23,32)(24,31)(26,29)(28,30);(17,18,21,24)(19,26,28,32)(20,29,27,23)(22,30,31,25)660 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 64);