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We find a sufficient condition to establish that certain abelian groups are not CI-groups with respect to ternary relational structures, and then show that the groups Z 3 ×Z 2 2 , Z 7 ×Z 3 2 , and Z 5 × Z 4 2 satisfy this condition. Then we completely determine which groups Z 3 2 × Z p , p a prime, are CI-groups with respect to color binary and ternary relational structures. Finally, we show that Z 5 2 is not a CI-group with respect to ternary relational structures.doi:10.26493/1855-3974.310.59f fatcat:6xxxrj75obhnzfnwgxb4eg3qti