Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13134 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C4×D8 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
324 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)(24,25,30,35,28,34,31,46)(26,36,39,53,38,32,29,44)(27,40,43,52,41,55,47,48)(33,45,50,37,49,54,51,42);(20,21,23,22)(24,26,31,29,28,38,30,39)(25,32,46,53,34,36,35,44)(27,37,47,45,41,42,43,54)(33,48,51,55,49,52,50,40);(20,22,23,21)(24,27,28,41)(25,33,34,49)(26,37,38,42)(29,45,39,54)(30,43,31,47)(32,48,36,52)(35,50,46,51)(40,53,55,44)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6122);