Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33469 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C2×#442 |
| commutator | C8 | C23×C4×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 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)(22,33,37,63)(23,39)(24,29)(25,46)(26,32)(27,36,52,71)(28,42)(30,44)(31,45)(34,64,67,83)(35,62,38,57)(40,54)(41,58)(43,55)(47,60,72,51)(48,53)(49,56)(50,76,81,79)(59,69)(61,68,80,65)(66,82,70,73)(74,77)(75,78);(20,22)(21,27)(23,34)(24,35)(25,47)(26,37)(28,50)(29,51)(30,57)(31,60)(32,52)(33,56)(36,69)(38,45)(39,73)(40,65)(41,76)(42,68)(43,67)(44,71)(46,63)(48,79)(49,72)(53,64)(54,81)(55,82)(58,83)(59,62)(61,74)(66,75)(70,77)(78,80);(20,23,24,42,26,43,45,40)(21,28,31,39,32,54,29,55)(22,34,35,68,37,67,38,65)(25,48,44,75,49,41,69,77)(27,50,60,73,52,81,51,82)(30,58,46,78,59,53,56,74)(33,61,57,83,63,80,62,64)(36,70,47,79,71,66,72,76);(20,24,26,45)(21,29,32,31)(22,35,37,38)(23,40,43,42)(25,44,49,69)(27,51,52,60)(28,39,54,55)(30,56,59,46)(33,62,63,57)(34,65,67,68)(36,47,71,72)(41,75,48,77)(50,73,81,82)(53,74,58,78)(61,83,80,64)(66,79,70,76)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26458);