Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#2222 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3×C5 |
| prime radical | C3×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C3×C5×#131 ◁ G | |
280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 960 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,3)(4,5,6,7,8)(9,10,16,22)(11,23,29,51)(12,30,33,47)(13,34,36,45)(14,21,15,20)(17,38,42,26)(18,37,44,32)(19,35,46,24)(25,54,57,71)(27,50,28,49)(31,62,63,43)(39,66,67,72)(40,58,41,52)(48,59,70,56)(53,61,64,69)(55,60,65,68);(9,11,24,52,15,28,36,38,16,29,35,58,14,27,13,26)(10,17,34,50,21,41,46,51,22,42,45,49,20,40,19,23)(12,25,53,72,32,56,63,65,33,57,64,66,37,59,31,55)(18,39,61,54,30,60,62,70,44,67,69,71,47,68,43,48);(9,12)(10,18)(11,25)(13,31)(14,37)(15,32)(16,33)(17,39)(19,43)(20,47)(21,30)(22,44)(23,48)(24,53)(26,55)(27,59)(28,56)(29,57)(34,61)(35,64)(36,63)(38,65)(40,68)(41,60)(42,67)(45,69)(46,62)(49,71)(50,54)(51,70)(52,72)(58,66)390 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 954);