The 4-GDDs of type 3562

Group divisible designs with block size 4 where the group sizes are congruent to 2 mod 3

“…The existence of 4-GDDs with no more than 30 points has been solved. In [13], existence was noted for all but three feasible group types here, namely types 2 3 5 4 , 3 5 6 2 and 2 2 5 5 . Those three types were solved in [2,3].…”

“…Theorem 1.2. [3,2,13] The only feasible group types for a 4-GDD on at most 30 points are 1 4 , 2 4 , 3 4 , 1 13 , 1 9 4 1 , 2 7 , 3 5 , 1 16 , 1 12 4 1 , 1 8 4 2 , 1 4 4 3 , 4 4 , 2 6 5 1 , 2 10 , 5 4 , 3 5 6 1 , 1 15 7 1 , 2 9 5 1 , 3 8 , 3 4 6 2 , 6 4 , 1 25 , 1 21 4 1 , 1 17 4 2 , 1 13 4 3 , 1 9 4 4 , 1 5 4 5 , 1 1 4 6 , 2 13 , 2 3 5 4 , 2 9 8 1 , 3 9 , 3 5 6 2 , 3 1 6 4 , 1 28 , 1 24 4 1 , 1 20 4 2 , 1 16 4 3 , 1 12 4 4 , 1 8 4 5 , 1 4 4 6 , 4 7 , 1 14 7 2 , 1 10 4 1 7 2 , 1 6 4 2 7 2 , 1 2 4 3 7 2 , 7 4 , 2 12 5 1 , 2 2 5 5 , 2 8 5 1 8 1 , 3 8 6 1 , 3 4 6 3 , 6 5 , 3 7 9 1 . A 4-GDD exists for all these types with the definite exception of types 2 4 , 2 6 5 1 and 6 4 .…”

Group divisible designs with block size 4 and group sizes 4 and 7

“…Other work on the existence of $$4$$‐GDDs has concentrated on whether the group sizes are congruent to $$0$$, $$1$$ or $$2\,(\mathrm{mod}\,3)$$ and on GDDs whose groups are of only two or three different sizes. See, for example, [1–7, 9, 10, 23].…”

“…Two of the $$4$$‐GDDs of type $${7}^{4}$$ have automorphism groups of order $$6$$ and the remaining ones have orders $$24$$, $$48$$, $$126$$, $$2352$$ and $$3528$$. In [6], the $$4$$‐GDDs of type $${3}^{5}{6}^{2}$$ were enumerated and only three out of $$22$$ had any nontrivial automorphisms. Of those that did have nontrivial automorphisms, one had an automorphism group of order $$2$$ and two had automorphism groups of order $$3$$.…”

“…In the past, several GDDs have been found directly by assuming the existence of a cyclic automorphism group, $${\mathbb{G}}$$. See for instance [1–6, 8–11, 22, 23]. In these designs, one assigns a selection of base blocks and then develops these base blocks to produce the blocks of the required design.…”