Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13098 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1898 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| prime radical | C19×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C19×2−1+4 ◁ G | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 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)(22,24,32,39)(23,33,36,45)(25,28)(26,29)(27,35,44,41)(30,47)(31,38,40,49)(34,48)(37,50)(42,46)(43,51);(20,22,24,29,26,32,39,21)(23,30,37,44,36,43,34,27)(25,40,38,28,42,31,49,46)(33,41,48,51,45,35,50,47);(20,23)(21,27)(22,30)(24,37)(25,41)(26,36)(28,45)(29,44)(31,50)(32,43)(33,46)(34,39)(35,42)(38,51)(40,48)(47,49)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6086);