Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13248 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1972 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
394 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)(22,34)(23,41,45,71)(24,47)(25,30)(26,51)(27,33)(28,37)(29,44,58,65)(31,49)(32,48)(35,60)(36,52,68,81)(38,54)(39,61)(40,64)(42,72,74,56)(43,75,77,59)(46,79,76,57)(50,63)(53,69)(55,70)(62,67,82,73)(66,80,78,83);(20,22,35,38,26,39,24,37,27,40,25,50,48,69,49,70)(21,28,31,61,32,54,30,34,33,55,47,53,51,63,60,64)(23,42,44,78,46,68,77,67,45,74,65,66,76,36,43,73)(29,56,71,82,59,81,79,83,58,72,41,62,75,52,57,80);(20,23,25,44,26,46,49,77,27,45,35,65,48,76,24,43)(21,29,31,41,32,59,30,57,33,58,47,71,51,75,60,79)(22,36,38,67,39,42,37,66,40,68,50,73,69,74,70,78)(28,52,53,80,54,56,64,82,55,81,61,83,63,72,34,62)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6236);