Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#33735 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C22×C19 | C22×D8 |
| prime radical | C19×#2215 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,25)(23,24)(26,27)(28,29)(30,35)(31,41)(32,46)(33,48)(34,38)(36,43)(37,42)(39,52)(40,51)(44,58)(45,55)(47,59)(49,53)(50,57)(54,56);(21,24)(25,27)(28,30,34,33)(29,35,38,48)(31,39,45,44)(32,40,47,42)(36,49,50,54)(37,46,51,59)(41,52,55,58)(43,53,57,56);(28,31,42,56,33,44,47,57,34,45,40,53,30,39,32,43)(29,36,46,52,35,49,51,55,38,50,59,58,48,54,37,41);(20,22,23,26)(21,25,24,27)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26723);