Let $G$ be the abelian group generated by elements $a, b, c, d$ subject to the relations

$$4 a-2 b+2 c+12 d=0, \quad-2 b+2 c=0, \quad 2 b+2 c=0, \quad 8 a+4 c+24 d=0$$

Express $G$ as a product of cyclic groups, and find the number of elements of $G$ of order 2 .