Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C27 |
| direct product | C19×#21009 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #386 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | C2×#386 |
| prime radical | C19×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
1708 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)(22,31)(23,36,38,28)(24,29)(25,39)(26,30)(27,34)(32,45,48,51)(33,42)(35,46)(37,44,40,43)(41,47,50,49);(20,22,25,34,26,35,24,33)(21,27,29,31,30,42,39,46)(23,32,40,49,38,48,37,47)(28,41,43,45,36,50,44,51);(20,23)(21,28)(22,32)(24,37)(25,40)(26,38)(27,41)(29,43)(30,36)(31,45)(33,47)(34,49)(35,48)(39,44)(42,50)(46,51);(20,24,26,25)(21,29,30,39)(22,33,35,34)(23,37,38,40)(27,31,42,46)(28,43,36,44)(32,47,48,49)(41,45,50,51)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13997);