Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11374 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #127 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #399 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
598 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)(20,21)(22,23)(24,25)(26,35,39,51)(27,40)(28,33)(29,34)(30,44)(31,37,47,38)(32,43)(36,52,53,48)(41,55,54,46)(42,50)(45,49);(21,23)(24,26,30,37,29,39,28,38)(25,31,44,35,34,47,33,51)(27,41,45,53,43,54,42,36)(32,48,50,46,40,52,49,55);(21,23)(24,27,30,45,29,43,28,42)(25,32,44,50,34,40,33,49)(26,36,37,41,39,53,38,54)(31,46,35,52,47,55,51,48)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4362);