Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#2175 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C5 | C2×D8 |
| commutator | C8 | C22×C3×C4×C5 |
| Frattini, cosocle | C2×C8 | C23×C3×C5 |
| prime radical | C2×C3×C5×D8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C5 ◁ C3×C42×C5 ◁ C3×C4×C5×C8 ◁ G | |
320 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)(4,5,6,7,8)(9,10,12,11)(13,14)(15,18)(16,25)(17,19)(20,22)(21,23)(24,26)(27,28);(14,18)(15,22)(16,23)(19,26)(20,21)(24,28)(25,27);(9,11,12,10)(14,19)(18,26)(20,27)(21,25)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 907);