Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C19 · Schur multiplier (H2) = C1 |
| direct product | C19×#7459 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C19 | D8 |
| commutator | C8 | C2×C16×C19 |
| Frattini, cosocle | C4×C16 | C22×C19 |
| socle | C2×C19 | #1370 |
| prime radical | C22×C19 | #364 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C19 ◁ C2×C16×C19 ◁ C4×C16×C19 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,28,23,31,36,45,26,30,41,44,38,47,37,50,27,32,42,46,39,48,35,51,25,43,40,49,34,29,22,33);(20,22,34,40,25,35,39,42,27,37,38,41,26,36,23,24)(21,28,31,45,30,44,47,50,32,46,48,51,43,49,29,33)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 447);