Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1874 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×#41×C19 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
472 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)(20,21,26,32,51,54,49,30,27,33,48,67,25,31,24,46)(22,34,38,28,50,57,36,65,39,69,78,58,37,63,52,56)(23,40,43,59,42,62,47,66,45,61,53,64,44,29,41,60)(35,70,73,79,72,80,71,55,75,82,77,81,74,68,76,83);(20,22,25,50,27,39,51,37)(21,28,54,56,33,58,31,65)(23,35,44,72,45,75,42,74)(24,38,48,36,49,78,26,52)(29,55,40,79,62,83,61,81)(30,63,67,69,46,57,32,34)(41,73,53,71,47,77,43,76)(59,70,66,68,64,82,60,80);(20,23)(21,29)(22,35)(24,47)(25,44)(26,53)(27,45)(28,55)(30,64)(31,61)(32,66)(33,62)(34,68)(36,76)(37,74)(38,77)(39,75)(40,54)(41,49)(42,51)(43,48)(46,59)(50,72)(52,71)(56,79)(57,70)(58,83)(60,67)(63,82)(65,81)(69,80)(73,78)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5994);