Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13249 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #442 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(25,31,35,34)(26,37,36,38)(27,32)(28,33)(29,39);(20,21,22,23)(24,25,32,38,39,26,27,34,30,35,33,37,29,36,28,31);(24,26,28,37,29,25,33,34,30,36,27,38,39,35,32,31)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6237);