Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#26839 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C22×D4×C19 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C4×C4⋊C4×C19 ◁ G | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25)(26,31)(27,29)(28,30)(32,33,37,41)(34,39,44,47)(35,45,43,38)(36,46,42,40);(20,21)(22,23)(24,26,28,27)(25,29,30,31)(32,34,42,35,37,44,36,43)(33,38,46,47,41,45,40,39);(21,23)(24,27,28,26)(25,29,30,31)(32,34,36,43,37,44,42,35)(33,38,40,39,41,45,46,47);(25,30)(29,31)(32,35,42,44,37,43,36,34)(33,39,46,38,41,47,40,45)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 19827);