Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12747 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #447 |
| 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 | |
886 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)(22,33)(23,38,41,56)(24,42)(25,31)(26,48)(27,32)(28,35)(29,40,55,47)(30,45)(34,62,65,82)(36,60)(37,61)(39,54,72,80)(43,51)(44,58)(46,74)(49,57)(50,77,78,70)(52,75)(53,67)(59,64,81,66)(63,68,83,79)(69,76,73,71);(20,22)(21,28)(23,34)(24,43)(25,36)(26,49)(27,37)(29,50)(30,57)(31,51)(32,53)(33,42)(35,45)(38,68)(39,70)(40,64)(41,65)(44,67)(46,75)(47,66)(48,52)(54,62)(55,78)(56,79)(58,60)(59,73)(61,74)(63,71)(69,81)(72,77)(76,83)(80,82);(20,23)(21,29)(22,34)(24,39)(25,47)(26,40)(27,41)(28,50)(30,54)(31,56)(32,55)(33,59)(35,63)(36,66)(37,65)(38,48)(42,73)(43,70)(44,76)(45,71)(46,72)(49,64)(51,79)(52,68)(53,78)(57,62)(58,80)(60,82)(61,81)(67,83)(69,74)(75,77)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5735);