Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C27 |
| direct product | C19×#20947 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×#120 |
| 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)(22,30,33,44)(23,34,36,27)(24,29)(25,28)(26,32,41,37)(31,46)(35,45,38,42)(39,43)(40,47)(48,51)(49,50);(20,22,24,37,25,33,39,32)(21,26,29,44,28,41,43,30)(23,31,38,47,36,48,35,49)(27,40,45,51,34,50,42,46);(20,23)(21,27)(22,31)(24,38)(25,36)(26,40)(28,34)(29,45)(30,46)(32,49)(33,48)(35,39)(37,47)(41,50)(42,43)(44,51);(21,28)(26,41)(27,34)(29,43)(30,44)(40,50)(42,45)(46,51)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13935);