Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#7475 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #367 |
| commutator | C16 | C42×C19 |
| Frattini, cosocle | C22×C16 | C22×C19 |
| socle | C2×C19 | #1380 |
| prime radical | C23×C19 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×C8×C19 ◁ C22×C16×C19 ◁ G | |
170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 2 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)(22,34,36,62)(23,30)(24,44)(25,47)(26,50)(27,33)(28,35,51,65)(29,41)(31,49)(32,48)(37,70,66,61)(38,72,68,83)(39,74,69,63)(40,57)(42,77)(43,60)(45,55)(46,56)(52,64,79,73)(53,81,80,71)(54,82,78,67)(58,76)(59,75);(20,22,24,37,45,68,75,82,48,69,76,81,49,73,43,65,27,36,41,66,25,38,42,67,26,39,46,71,40,64,23,35)(21,28,30,52,57,80,56,63,50,78,77,83,47,70,29,34,33,51,60,79,31,53,58,74,32,54,59,72,55,61,44,62)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 463);