Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11320 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #397 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,22)(24,25)(26,34,37,47)(27,31)(28,32)(29,33)(30,36,44,41)(35,50,38,49)(39,45)(40,46)(42,48)(43,53,54,52)(51,55);(20,21)(22,23)(24,26,28,41,29,37,42,36)(25,30,32,47,33,44,48,34)(27,38,39,52,40,35,51,53)(31,43,45,50,46,54,55,49);(20,22,23,21)(24,27,28,39,29,40,42,51)(25,31,32,45,33,46,48,55)(26,35,36,52,37,38,41,53)(30,43,34,49,44,54,47,50)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4308);