Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#33693 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| prime radical | C19×#575 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C19×#405 ◁ G | |
512 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)(20,21,26,31)(22,27,33,49)(23,36,38,71)(24,41,44,76)(25,47,45,30)(28,52,54,39)(29,56,59,46)(32,60,62,70)(34,65,42,75)(35,68,64,51)(37,58,40,74)(43,53,72,55)(48,73,80,63)(50,78,57,67)(61,83,69,81)(66,82,77,79);(20,22,26,33)(21,27,31,49)(23,32,38,62)(24,42,44,34)(25,35,45,64)(28,48,54,80)(29,57,59,50)(30,51,47,68)(36,70,71,60)(37,69,40,61)(39,73,52,63)(41,75,76,65)(43,77,72,66)(46,78,56,67)(53,82,55,79)(58,83,74,81);(20,23)(21,28)(22,32)(24,43)(25,37)(26,38)(27,48)(29,58)(30,53)(31,54)(33,62)(34,66)(35,69)(36,41)(39,46)(40,45)(42,77)(44,72)(47,55)(49,80)(50,81)(51,82)(52,56)(57,83)(59,74)(60,65)(61,64)(63,67)(68,79)(70,75)(71,76)(73,78);(22,33)(23,37,38,40)(24,44)(27,49)(28,53,54,55)(29,59)(32,61,62,69)(35,64)(36,58,71,74)(39,43,52,72)(41,76)(46,56)(48,79,80,82)(51,68)(60,83,70,81)(63,77,73,66)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26681);