Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12947 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1862 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,26,33,28,34,27,49)(22,35,39,65,40,66,50,53)(23,41,44,62,45,77,51,57)(24,32,46,61,47,63,25,31)(29,38,54,73,56,75,55,37)(30,42,58,79,60,48,59,43)(36,67,70,81,72,82,71,52)(64,74,80,78,83,68,76,69);(20,22,26,39,28,40,27,50)(21,29,33,54,34,56,49,55)(23,36,44,70,45,72,51,71)(24,38,46,73,47,75,25,37)(30,52,58,67,60,81,59,82)(31,53,32,35,61,65,63,66)(41,76,62,64,77,80,57,83)(42,78,79,68,48,69,43,74);(20,23)(21,30)(22,36)(24,43)(25,48)(26,44)(27,51)(28,45)(29,52)(31,41)(32,62)(33,58)(34,60)(35,64)(37,69)(38,74)(39,70)(40,72)(42,46)(47,79)(49,59)(50,71)(53,76)(54,67)(55,82)(56,81)(57,63)(61,77)(65,80)(66,83)(68,75)(73,78)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5935);