Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C2×C42 |
| direct product | C19×#12622 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#99 |
| commutator | C23 | C2×C42×C19 |
| Frattini, cosocle | C22×D4 | C23×C19 |
| prime radical | C2×C19×#120 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C4×D4×C19 ◁ G | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(24,25,28,30)(26,32,34,39)(27,31,36,38)(29,35,37,33);(24,26)(25,29)(27,35,36,33)(28,34)(30,37)(31,39,38,32);(20,22,23,21)(25,30)(29,37)(31,38)(32,39)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5610);