Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11119 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #394 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C2×C19 | #1439 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
864 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)(22,32)(23,36,39,28)(24,31)(25,29)(26,30)(27,34)(33,48,50,44)(35,46)(37,51,49,42)(38,45,41,47)(40,43);(20,22,24,34,26,35,25,40)(21,27,29,32,30,43,31,46)(23,37,38,33,39,49,41,50)(28,44,45,42,36,48,47,51);(20,23)(21,28)(22,33)(24,38)(25,41)(26,39)(27,42)(29,45)(30,36)(31,47)(32,48)(34,49)(35,50)(37,40)(43,51)(44,46)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4107);