Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C2×C19×#2104 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #401 |
| commutator, Frattini, cosocle | C24 | C24×C19 |
| prime radical | C25×C19 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C25×C19 ◁ G | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(3,4)(5,11,13,8)(6,9,15,18)(7,10)(12,17)(14,16);(3,5)(4,8)(6,14,15,12)(7,13)(9,17,18,16)(10,11);(3,6)(4,9,10,18)(5,12)(7,15)(8,16,11,17)(13,14)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16834);