Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#26730 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C19×#1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C4×C22⋊C4×C19 ◁ G | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
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)(22,25)(23,24)(26,27)(28,29)(30,33)(31,32)(34,35)(36,37,39,42)(38,44,46,41)(40,43,48,51)(45,50,47,49);(21,24)(25,27)(36,38)(37,41)(39,46)(40,45)(42,44)(43,49)(47,48)(50,51);(20,22,23,26)(21,25,24,27)(29,32)(33,35)(38,45)(40,48)(41,49)(43,51)(44,50)(46,47);(21,24)(25,27)(28,30,31,34)(29,33,32,35)(37,42)(41,44)(43,51)(49,50)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 19718);