The doubly truncated Witt graph is the graph on 330 vertices related to a 3-
design (Brouwer et al. 1989, p. 367).
The doubly truncated Witt graph can be constructed by removing all vectors of the large Witt design containing any two arbitrarily chosen symbols. Consider the 759 vertices of the large Witt graph as words of weight 8 in extended binary Golay. Call the octads, and view them as sets of size 8. Pick one coordinate position. 253 octads have a 1 there, 506 octads have a 0 there. Pick a second coordinate position. Of the 506, there are 176 with a 1 there, and 330 with a 0. The induced subgraph from 759, or from 506 on this 330 gives the doubly truncated Witt graph (A. E. Brouwer, pers. comm., Jun. 8, 2009).
It is an integral graph with graph spectrum 
,
is weakly regular with parameters 
. It is also distance-transitive
with intersection array 
. The order of its automorphism
group is 
,
where 
is a Mathieu group.
This graph is implemented in the Wolfram Language as GraphData["DoublyTruncatedWittGraph"].
Van Dam and Haemers (2003) designate this graph as 
, but it is distinct from the M22
graph.
." §11.4C in