Given a graph $G$ and the set $C_k(G)$ of the $k$-cliques in $G$, one can build a clique graph $H$ whose vertices $c_i\in C_k(G)$ are connected if the vertex sets of $c_i$ and $c_j$ have an intersection of cardinality $k-1$.
This construction is useful when one is interested in clique percolation.
My question is the following: when is $H\simeq G$ ?
An example:
$G$ formed by the 3-cliques $\{1,2,3\}$ and $\{4,5,6\}$, as well as the 4 clique $\{2,3,4,5\}$.
The set of 3 cliques is $c_1=\{1,2,3\}$, $c_2=\{2,3,4\}$, $c_3=\{2,3,5\}$, $c_4=\{3,4,5\}$, $c_5=\{2,4,5\}$ and $c_6=\{4,5,6\}$.
One can verify that $H$ has the following edge set: $\{(c_1,c_2), (c_1,c_3), (c_2,c_3), (c_2,c_4), (c_2,c_5), (c_3,c_4), (c_3,c_5), (c_4,c_5), (c_4,c_6),(c_5,c_6)\}$, i.e. it is isomorphic to $G$.
Is this graph special?