Title: Automorphism of a graph which is like Kneser graph
Tags: gr.group-theory;graph-theory;automorphism-groups
Question: Thanks for any help or comment.
Consider the following graph:
$X=\{1,2,\dots,6\}$ and a vertex of graph is the following set $Y=\{a,b\}\cup\{c,d\}$, where $a,b,c,d\in X$. There exist an edge between to vertices $Y_1=\{a,b\}\cup\{c,d\}$ and $Y_2=\{a',b'\}\cup\{c',d'\}$ if and only if $Y_1\cap Y_2=\{a,b\}$ or $\{c,d\}$. So this graph has 45 vertices and for example if $Z_1=\{1,2\}\cup\{3,4\}$ and $Z_2=\{1,3\}\cup\{2,4\}$ and $Z_3=\{1,2\}\cup\{5,6\}$, then there exist an edge between $Z_1,Z_3$ and there is no edge between
$Z_1,Z_2$ and $Z_2,Z_3$.
What is the structure of automorphism group of this graph?
Comment: It is possible to construct the graph using software packages such as SAGE or GAP.  They have built-in functions to compute the automorphism group and describe its structure.

Comment: I put it into SAGE for you. The automorphism group is $S_6$, induced by the action of $S_6$ on $X$.



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