Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12833 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12833 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1807 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×#41×C19 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
358 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)(22,33,37,51)(23,38,40,48)(24,34)(25,30)(26,32)(27,35,41,28)(29,42)(31,44)(36,50,45,39)(43,46)(47,49);(20,22,24,35,25,36,42,48,26,37,43,28,44,45,49,38)(21,27,29,33,31,23,34,39,32,41,47,51,30,40,46,50);(20,23)(21,28)(22,34)(24,39)(25,41)(26,40)(27,44)(29,45)(30,48)(31,38)(32,35)(33,49)(36,47)(37,46)(42,51)(43,50)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5821);