Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#26231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C22×D4×C19 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C4×D4×C19 ◁ G | |
862 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)(21,22)(24,25)(26,30)(27,28)(29,31)(32,33,36,39)(34,37,42,46)(35,38,43,47)(40,44,41,45);(20,21,23,22)(25,28)(29,30)(32,34)(33,37)(35,40)(36,42)(38,44)(39,46)(41,43)(45,47);(20,21,23,22)(34,40)(35,43)(37,44)(38,47)(41,42)(45,46);(20,22,23,21)(24,26,27,31)(25,29,28,30)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 19219);