Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11212 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #386 |
| prime radical | C24×C19 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
714 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)(21,22)(24,25,28,32,29,33,27,36)(26,30,34,31,35,39,37,38);(20,21,23,22)(24,26)(25,30)(27,37)(28,34)(29,35)(31,32)(33,39)(36,38);(20,22,23,21)(24,26)(25,31)(27,37)(28,34)(29,35)(30,36)(32,39)(33,38)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4200);