Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C23 |
| direct product | C3×C5×#2195 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3×C5 |
| prime radical | C3×C5×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C3×C5×#131 ◁ G | |
412 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)(11,23,27,51)(12,28,31,48)(13,32,36,46)(14,21,15,20)(17,39,42,25)(18,37,44,30)(19,34,47,35)(24,52,56,69)(26,57,38,45)(29,60,63,43)(33,49,64,41)(40,61,68,72)(50,55,71,54)(53,58,66,67)(59,65,70,62);(9,11)(10,17)(12,24)(13,33)(14,26)(15,38)(16,27)(18,40)(19,45)(20,41)(21,49)(22,42)(23,32)(25,35)(28,58)(29,61)(30,54)(31,56)(34,39)(36,64)(37,55)(43,69)(44,68)(46,51)(47,57)(48,67)(50,59)(52,60)(53,62)(63,72)(65,66)(70,71);(9,12,16,31)(10,18,22,44)(11,24,27,56)(13,29,36,63)(14,37,15,30)(17,40,42,68)(19,43,47,60)(20,48,21,28)(23,50,51,71)(25,53,39,66)(26,55,38,54)(32,59,46,70)(33,61,64,72)(34,65,35,62)(41,67,49,58)(45,69,57,52)345 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 927);