Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33441 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C19 | C2×D4 |
| commutator | C4 | C23×C8×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C2×C19 | C2×C8×D4 |
| prime radical | C19×#575 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C19 ◁ C22×C16×C19 ◁ G | |
646 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)(20,21,25,32,26,31,45,56,27,30,46,55,24,42,44,68)(22,33,36,58,35,28,47,50,37,51,62,75,43,49,69,74)(23,29,39,54,40,53,66,77,41,52,67,76,38,63,65,80)(34,57,60,78,59,48,70,72,61,73,79,83,64,71,81,82);(20,22)(21,28)(23,34)(24,43)(25,36)(26,35)(27,37)(29,48)(30,49)(31,51)(32,50)(33,42)(38,64)(39,60)(40,59)(41,61)(44,69)(45,47)(46,62)(52,71)(53,73)(54,72)(55,74)(56,75)(57,63)(58,68)(65,81)(66,70)(67,79)(76,82)(77,83)(78,80);(20,23,25,39,26,40,45,66,27,41,46,67,24,38,44,65)(21,29,32,54,31,53,56,77,30,52,55,76,42,63,68,80)(22,34,36,60,35,59,47,70,37,61,62,79,43,64,69,81)(28,48,50,72,51,73,75,83,49,71,74,82,33,57,58,78);(21,30)(28,49)(29,52)(31,42)(32,55)(33,51)(48,71)(50,74)(53,63)(54,76)(56,68)(57,73)(58,75)(72,82)(77,80)(78,83)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26429);