Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C4 |
| direct product | C19×#13115 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #481 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1919 |
| prime radical | C23×C19 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
252 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)(20,21,26,31,24,30,27,41)(22,32,35,49,34,28,25,39)(23,36,37,43,38,51,42,48)(29,44,45,50,47,40,46,33);(20,22,26,35,24,34,27,25)(21,28,31,39,30,32,41,49)(23,33,37,44,38,50,42,40)(29,43,45,51,47,48,46,36);(20,23,26,37,24,38,27,42)(21,29,41,46,30,47,31,45)(22,33,35,44,34,50,25,40)(28,43,49,36,32,48,39,51)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6103);