Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13232 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1964 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| prime radical | C19×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C19×#131 ◁ G | |
448 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)(20,21,26,31,27,32,25,44)(22,33,37,40,38,48,36,51)(23,39,28,46,42,35,41,29)(24,30,34,47,43,50,45,49);(20,22,34,29)(21,28,47,40)(23,30,48,31)(24,35,26,37)(25,36,43,39)(27,38,45,46)(32,41,49,51)(33,44,42,50);(20,23,25,41,27,42,26,28)(21,29,31,39,32,46,44,35)(22,30,36,49,38,50,37,47)(24,40,34,48,43,51,45,33)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6220);