Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13158 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1936 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C22×D8×C19 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,26,31,28,32,27,42)(22,24,30,43,35,33,45,25)(23,36,40,51,41,37,46,34)(29,47,48,39,50,38,49,44);(20,22,33,31,27,30,43,32,28,35,24,42,26,45,25,21)(23,37,38,29,40,51,44,49,41,36,47,50,46,34,39,48);(20,23,24,38,27,40,25,44,28,41,33,47,26,46,43,39)(21,29,22,34,31,49,30,37,32,50,35,51,42,48,45,36)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6146);