Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#8157 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | C4×C22⋊C4 |
| prime radical | C24×C19 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(28,29,32,36,33,37,44,50)(30,38,41,52,40,34,45,47)(31,35,42,48,43,49,55,58)(39,51,54,59,53,46,56,57);(20,21,22,25,23,24,26,27)(28,30)(29,34,37,38)(31,39)(32,41)(33,40)(35,46,49,51)(36,47,50,52)(42,54)(43,53)(44,45)(48,57,58,59)(55,56);(21,24)(25,27)(28,31,32,42,33,43,44,55)(29,35,36,48,37,49,50,58)(30,39,41,54,40,53,45,56)(34,46,47,57,38,51,52,59)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1145);