Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#19024 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#3558 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 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)(22,30)(23,27)(24,28)(25,29)(26,31)(32,34)(33,35)(36,37,41,44)(38,45,48,51)(39,43,40,49)(42,46,50,47);(20,22)(21,26)(23,33)(24,31)(25,32)(27,30)(28,34)(29,35)(36,38)(37,42)(39,47)(40,46)(41,48)(43,51)(44,50)(45,49);(20,22)(21,26)(23,31)(24,33)(25,32)(27,34)(28,30)(29,35)(36,37)(38,42)(39,43)(40,49)(41,44)(45,46)(47,51)(48,50);(20,23,25,24)(21,27,29,28)(22,31,32,33)(26,34,35,30)(36,39,41,40)(37,43,44,49)(38,46,48,47)(42,45,50,51)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12012);