Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C3×C5×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | C2×#99 |
| commutator | C2×C4 | C22×C3×C4×C5 |
| Frattini, cosocle | C2×Q8 | C23×C3×C5 |
| prime radical | C2×C3×C5×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C23×C3×C5 ◁ C22×C3×C5×Q8 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 480 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3)(4,5,6,7,8)(9,10,13,17,14,18,29,36)(11,19,22,37,20,33,27,39)(12,23,25,35,26,16,28,34)(15,30,32,24,31,38,40,21);(9,11)(10,15)(12,24)(13,27)(14,20)(16,33)(17,32)(18,31)(19,23)(21,26)(22,29)(25,30)(28,38)(34,39)(35,37)(36,40);(11,20)(15,31)(19,33)(21,24)(22,27)(30,38)(32,40)(37,39)345 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 865);