Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#1495 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C42 | C23×C3×C5 |
| socle | C22×C3×C5 | C2×C22⋊C4 |
| prime radical | C3×C5×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ C2×C3×C42×C5 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 960 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3)(4,5,6,7,8)(10,13)(11,15)(14,16)(17,18,20,24,21,23,27,30)(19,22,26,28,25,29,31,32);(9,10)(11,16)(12,14)(13,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(10,14)(13,16)(18,23)(19,25)(24,30)(26,31)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 227);