Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1936 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
306 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)(21,27,31,38)(22,33,36,41)(23,28)(24,39)(25,42)(29,44)(30,47,40,32)(35,49,45,37)(43,48)(46,50);(20,21,28,30,42,47,23,38,26,31,24,40,25,32,39,27)(22,34,41,50,49,29,45,46,36,51,33,43,37,44,35,48);(20,22,23,35,25,37,28,33,26,36,39,45,42,49,24,41)(21,29,27,43,32,34,40,46,31,44,38,50,47,51,30,48)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6144);