Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C4×C5×SD32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | D8 |
| commutator | C8 | C22×C3×C4×C5 |
| Frattini, cosocle | C2×C8 | C23×C3×C5 |
| socle | C22×C3×C5 | C2×D8 |
| prime radical | C2×C3×C5×D8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ C3×C42×C5 ◁ C3×C4×C5×C8 ◁ G | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 960 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3)(4,5,6,7)(24,25,26,27,28);(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23);(9,15)(10,22)(11,13)(12,20)(14,18)(17,23)(19,21)660 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 905);