Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #442 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,22)(24,25)(26,36)(27,39,41,47)(28,42)(29,31)(30,35)(32,37,50,55)(33,44)(34,38)(40,52,43,53)(45,51,54,49)(46,48);(20,21,23,22)(24,26,28,29,30,38,44,46)(25,31,33,36,35,48,42,34)(27,37,43,45,41,55,40,54)(32,47,51,53,50,39,49,52);(20,22,23,21)(24,27,30,41)(25,32,35,50)(26,37,38,55)(28,43,44,40)(29,45,46,54)(31,47,48,39)(33,51,42,49)(34,52,36,53)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6129);