Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#20324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#555 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C42×C19 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,23)(22,28,31,35)(24,33,34,27)(26,29,32,30)(36,37,41,44)(38,42,40,49)(39,46,48,50)(43,47,51,45);(20,22,25,31)(21,26,23,32)(24,29,34,30)(27,35,33,28)(36,38,41,40)(37,42,44,49)(39,47,48,45)(43,46,51,50);(20,23,25,21)(22,26,31,32)(24,33,34,27)(28,29,35,30)(36,39,41,48)(37,43,44,51)(38,45,40,47)(42,50,49,46);(22,29)(24,34)(26,35)(27,33)(28,32)(30,31)(37,44)(42,49)(43,51)(46,50)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13312);