Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#19028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#3217 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
888 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)(21,26)(22,29)(23,24)(28,31)(30,32)(33,35)(37,41)(39,46)(42,50)(43,47)(44,48)(49,51);(20,21)(22,27)(23,31)(24,26)(25,28)(29,35)(30,34)(32,33)(36,37,40,43)(38,42,45,51)(39,47,46,41)(44,49,48,50);(20,22,25,30)(21,27,28,34)(23,29,24,32)(26,33,31,35)(36,38,40,45)(37,42,43,51)(39,48,46,44)(41,49,47,50);(21,28)(26,31)(27,34)(33,35)(37,43)(41,47)(42,51)(49,50)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12016);