Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#22695 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | D42 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | D4×C4⋊C4 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ G | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
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)(20,21,25,32,27,33,26,31)(22,34,38,46,39,50,37,28)(23,40,41,48,42,45,36,29)(24,30,44,49,35,47,43,51);(20,22)(21,28)(23,24)(25,38)(26,37)(27,39)(29,47)(30,48)(31,50)(32,34)(33,46)(35,42)(36,43)(40,51)(41,44)(45,49);(20,23,26,36,27,42,25,41)(21,29,32,40,33,48,31,45)(22,35,37,44,39,24,38,43)(28,30,34,49,46,47,50,51);(20,24)(21,30)(22,36)(23,38)(25,44)(26,43)(27,35)(28,45)(29,34)(31,51)(32,49)(33,47)(37,42)(39,41)(40,46)(48,50)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 15683);