Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#23047 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | D42 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C2×#41×C19 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ G | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(24,25,29,34,30,35,28,33)(26,36,39,31,40,47,38,48)(27,41,44,50,45,32,43,51)(37,52,53,55,54,49,42,46);(20,22,23,21)(24,26)(25,31)(27,42)(28,39)(29,38)(30,40)(32,49)(33,47)(34,36)(35,48)(37,43)(41,52)(44,54)(45,53)(46,50)(51,55);(24,26)(25,31)(27,37)(28,38)(29,39)(30,40)(32,46)(33,36)(34,47)(35,48)(41,55)(42,43)(44,53)(45,54)(49,50)(51,52);(24,27,28,43,30,45,29,44)(25,32,33,50,35,41,34,51)(26,37,38,42,40,54,39,53)(31,46,36,49,48,55,47,52)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16035);