Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12744 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1786 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
874 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)(20,21,25,32,27,31,26,48)(22,33,36,61,37,60,49,57)(23,38,40,56,41,29,47,55)(24,42,44,58,46,75,45,30)(28,43,52,67,51,76,53,35)(34,62,64,81,66,59,65,80)(39,70,72,69,73,74,77,54)(50,71,78,63,68,82,79,83);(20,22)(21,28)(23,34)(24,43)(25,36)(26,49)(27,37)(29,50)(30,57)(31,51)(32,52)(33,42)(35,45)(38,68)(39,71)(40,64)(41,66)(44,67)(46,76)(47,65)(48,53)(54,80)(55,78)(56,79)(58,61)(59,74)(60,75)(62,70)(63,72)(69,81)(73,82)(77,83);(20,23,27,41)(21,29,31,38)(22,34,37,66)(24,39,46,73)(25,47,26,40)(28,50,51,68)(30,54,58,69)(32,56,48,55)(33,59,60,62)(35,63,67,83)(36,65,49,64)(42,74,75,70)(43,71,76,82)(44,77,45,72)(52,79,53,78)(57,80,61,81)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5732);