Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#23601 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×#121 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
994 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)(24,25,28,33,30,34,29,43)(26,35,37,46,39,52,38,31)(27,32,44,49,42,50,41,48)(36,51,54,45,53,47,40,55);(20,21)(22,23)(24,26,29,38,30,39,28,37)(25,31,33,35,34,46,43,52)(27,40,41,53,42,54,44,36)(32,47,49,55,50,51,48,45);(21,23)(24,27)(25,32)(26,36)(28,41)(29,44)(30,42)(31,45)(33,48)(34,50)(35,51)(37,40)(38,54)(39,53)(43,49)(46,55)(47,52);(24,28,30,29)(25,33,34,43)(26,37,39,38)(27,41,42,44)(31,35,46,52)(32,48,50,49)(36,40,53,54)(45,51,55,47)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16589);