Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C27 |
| direct product | C19×#20964 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×#120 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | C2×#386 |
| prime radical | C19×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
1700 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)(20,21,24,29,25,28,37,42)(22,26,32,40,31,39,46,50)(23,33,35,43,34,27,36,41)(30,38,45,49,44,48,47,51);(20,22,24,32,25,31,37,46)(21,26,29,40,28,39,42,50)(23,30,36,47,34,44,35,45)(27,38,43,51,33,48,41,49);(20,23)(21,27)(22,30)(24,36)(25,34)(26,38)(28,33)(29,43)(31,44)(32,47)(35,37)(39,48)(40,51)(41,42)(45,46)(49,50);(21,28)(26,39)(27,33)(29,42)(38,48)(40,50)(41,43)(49,51)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13952);