Finite Group DatabasebetaFinite Group Databasebeta| order | 1744 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C109 · Schur multiplier (H2) = C1 |
| direct product | Q16×C109 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C109 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C109 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C4×C109 ◁ G | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 1744 |
A minimal faithful permutation representation — 2 generators on 125 points:
(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)763 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1744, 9);