Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12745 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #418 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1786 |
| prime radical | C19×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
890 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)(22,32)(23,37,39,28)(24,41)(25,47)(26,30)(27,34)(29,45)(31,44)(33,58,61,79)(35,59)(36,65)(38,68,43,76)(40,53,70,67)(42,57)(46,73)(48,54)(49,60,66,63)(50,75)(51,64)(52,71,55,77)(56,78)(62,81,72,83)(69,80,74,82);(20,22)(21,27)(23,33)(24,42)(25,48)(26,35)(28,49)(29,54)(30,51)(31,57)(32,41)(34,45)(36,44)(37,66)(38,69)(39,61)(40,72)(43,74)(46,75)(47,50)(52,81)(53,82)(55,83)(56,65)(58,68)(59,73)(60,71)(62,70)(63,77)(64,78)(67,80)(76,79);(20,23)(21,28)(22,33)(24,43)(25,40)(26,39)(27,49)(29,55)(30,37)(31,53)(32,58)(34,63)(35,61)(36,62)(38,46)(41,68)(42,74)(44,70)(45,77)(47,67)(48,72)(50,80)(51,66)(52,56)(54,83)(57,82)(59,79)(60,64)(65,81)(69,75)(71,78)(73,76)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5733);