Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33465 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C22×D8 |
| commutator | C8 | C23×C4×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C2×C19 | C2×C4×D8 |
| prime radical | C2×C19×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×M16×C19 ◁ C19×#405 ◁ G | |
642 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)(20,21,25,32,27,30,26,47)(22,33,36,60,37,59,48,57)(23,29,40,54,38,55,41,70)(24,42,44,56,46,72,45,31)(28,43,51,64,50,73,52,35)(34,58,63,79,61,80,71,77)(39,66,68,78,65,82,69,53)(49,67,74,81,76,83,75,62);(20,22)(21,28)(23,34)(24,43)(25,36)(26,48)(27,37)(29,49)(30,50)(31,57)(32,51)(33,42)(35,45)(38,61)(39,67)(40,63)(41,71)(44,64)(46,73)(47,52)(53,77)(54,74)(55,76)(56,60)(58,66)(59,72)(62,69)(65,83)(68,81)(70,75)(78,79)(80,82);(20,23,25,40,27,38,26,41)(21,29,32,54,30,55,47,70)(22,34,36,63,37,61,48,71)(24,39,44,68,46,65,45,69)(28,49,51,74,50,76,52,75)(31,53,42,66,56,78,72,82)(33,58,60,79,59,80,57,77)(35,62,43,67,64,81,73,83);(21,30)(23,38)(28,50)(31,56)(32,47)(33,59)(34,61)(39,65)(40,41)(42,72)(51,52)(57,60)(62,81)(63,71)(67,83)(68,69)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26453);