> < ^ Date: Tue, 18 Nov 1997 13:34:48 GMT
> < ^ From: Leonard Soicher <l.h.soicher@qmul.ac.uk >
< ^ Subject: Re: Graph of 3.Sym(7)

Dear GAP-Forum,

Dear GAP Forum,

How does one construct the graph of 3.Sym(7) in GAP?

Ed Cureg

By the graph of 3.Sym(7) I take you to mean the "Conway-Smith graph for
3.Sym(7)" described in Theorem 13.2.3, p. 399, of [BCN] (Brouwer, Cohen,
and Neumaier, Distance-regular Graphs, Springer, 1989). If this is not
the case then please tell me.

One should be able to construct this graph E directly in GRAPE using
the function "Graph" and the description of E in [BCN]. However, I
have chosen to construct E as the the 1-skeleton of the universal cover
of the clique complex of the complement J of the Johnson graph J(7,2).
J has vertex-set the 21 2-subsets of {1,...,7} with vertices v and w
adjacent iff they have trivial intersection. The functions
FundamentalRecCliqueComplex and CoveringGraph I use are not yet
part of GRAPE, but are part of a new suite of algorithms and programs for
fundamental groups and covers being developed by Sarah Rees and myself.
Note that the graph E (I assume) you want is called U in the logfile
below. Note how the vertices of U are named, and that thre group associated
with U is 3.Sym(7).

Hope this helps.

Regards, Leonard

```--------------------------------------------------------------------------
```

gap> RequirePackage("grape");

by L.H.Soicher@qmw.ac.uk.

```gap> Read("gapprogs/homolo.new");
gap> J:=ComplementGraph(JohnsonGraph(7,2));;
gap> F:=FundamentalRecCliqueComplex(J);;
gap> Size(F.group);
3
gap> # this shows that the fundamental group of the clique
gap> # complex of  J  has order  3.  We now construct the
gap> # 1-skeleton of the universal cover of this complex.
gap>
gap> U:=CoveringGraph(J,F.group,F.edgeLabels,TrivialSubgroup(F.group));
rec(
isGraph := true,
order := 63,
group := Group( ( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)
(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)
(40,41,42)(43,44,45)(46,47,48)(49,50,51)(52,53,54)(55,56,57)(58,59,60)
(61,62,63), ( 1,31)( 2,33)( 3,32)( 4,44)( 5,43)( 6,45)( 7,54)( 8,53)
( 9,52)(10,58)(11,60)(12,59)(13,61)(14,63)(15,62)(17,18)(19,20)(22,24)
(26,27)(29,30)(35,36)(37,39)(40,41)(46,47)(49,51)(56,57), ( 1,16)( 2,18)
( 3,17)( 4, 6)( 7, 8)(11,12)(14,15)(19,45)(20,44)(21,43)(22,53)(23,52)
(24,54)(25,58)(26,60)(27,59)(28,61)(29,63)(30,62)(32,33)(35,36)(37,38)
(40,42)(46,48)(49,50)(56,57), ( 2, 3)( 4,17)( 5,16)( 6,18)( 7, 8)(10,12)
(13,14)(19,33)(20,32)(21,31)(22,24)(25,26)(28,30)(34,52)(35,54)(36,53)
(37,58)(38,60)(39,59)(40,61)(41,63)(42,62)(44,45)(47,48)(50,51)(56,57),
( 2, 3)( 4, 6)( 7,18)( 8,17)( 9,16)(10,11)(13,15)(19,20)(22,32)(23,31)
(24,33)(25,27)(28,29)(34,43)(35,45)(36,44)(38,39)(41,42)(46,58)(47,60)
(48,59)(49,61)(50,63)(51,62)(53,54)(56,57), ( 2, 3)( 4, 5)( 7, 9)(10,16)
(11,18)(12,17)(14,15)(19,21)(22,23)(25,31)(26,33)(27,32)(29,30)(35,36)
(37,43)(38,45)(39,44)(41,42)(46,52)(47,54)(48,53)(50,51)(55,61)(56,63)
(57,62)(59,60), ( 2, 3)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17)(20,21)
(23,24)(26,27)(28,31)(29,33)(30,32)(35,36)(38,39)(40,43)(41,45)(42,44)
(47,48)(49,52)(50,54)(51,53)(55,58)(56,60)(57,59)(62,63) ),
schreierVector := [ -1, 1, 1, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 3, 1, 3,
4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 2, 1, 2, 5, 5, 5, 6, 6, 6, 7, 7, 7,
3, 3, 3, 6, 6, 6, 7, 7, 7, 3, 3, 3, 7, 7, 7, 3, 3, 3, 3, 3, 3 ],
adjacencies := [ [ 34, 37, 40, 43, 46, 49, 52, 55, 58, 61 ] ],
representatives := [ 1 ],
isSimple := true,
names := [ [ [ 1, 2 ], 1 ], [ [ 1, 2 ], 2 ], [ [ 1, 2 ], 3 ],
[ [ 1, 3 ], 1 ], [ [ 1, 3 ], 2 ], [ [ 1, 3 ], 3 ], [ [ 1, 4 ], 1 ],
[ [ 1, 4 ], 2 ], [ [ 1, 4 ], 3 ], [ [ 1, 5 ], 1 ], [ [ 1, 5 ], 2 ],
[ [ 1, 5 ], 3 ], [ [ 1, 6 ], 1 ], [ [ 1, 6 ], 2 ], [ [ 1, 6 ], 3 ],
[ [ 1, 7 ], 1 ], [ [ 1, 7 ], 2 ], [ [ 1, 7 ], 3 ], [ [ 2, 3 ], 1 ],
[ [ 2, 3 ], 2 ], [ [ 2, 3 ], 3 ], [ [ 2, 4 ], 1 ], [ [ 2, 4 ], 2 ],
[ [ 2, 4 ], 3 ], [ [ 2, 5 ], 1 ], [ [ 2, 5 ], 2 ], [ [ 2, 5 ], 3 ],
[ [ 2, 6 ], 1 ], [ [ 2, 6 ], 2 ], [ [ 2, 6 ], 3 ], [ [ 2, 7 ], 1 ],
[ [ 2, 7 ], 2 ], [ [ 2, 7 ], 3 ], [ [ 3, 4 ], 1 ], [ [ 3, 4 ], 2 ],
[ [ 3, 4 ], 3 ], [ [ 3, 5 ], 1 ], [ [ 3, 5 ], 2 ], [ [ 3, 5 ], 3 ],
[ [ 3, 6 ], 1 ], [ [ 3, 6 ], 2 ], [ [ 3, 6 ], 3 ], [ [ 3, 7 ], 1 ],
[ [ 3, 7 ], 2 ], [ [ 3, 7 ], 3 ], [ [ 4, 5 ], 1 ], [ [ 4, 5 ], 2 ],
[ [ 4, 5 ], 3 ], [ [ 4, 6 ], 1 ], [ [ 4, 6 ], 2 ], [ [ 4, 6 ], 3 ],
[ [ 4, 7 ], 1 ], [ [ 4, 7 ], 2 ], [ [ 4, 7 ], 3 ], [ [ 5, 6 ], 1 ],
[ [ 5, 6 ], 2 ], [ [ 5, 6 ], 3 ], [ [ 5, 7 ], 1 ], [ [ 5, 7 ], 2 ],
[ [ 5, 7 ], 3 ], [ [ 6, 7 ], 1 ], [ [ 6, 7 ], 2 ], [ [ 6, 7 ], 3 ] ] )
gap> Size(U.group);
15120
gap> GlobalParameters(U);
[ [ 0, 0, 10 ], [ 1, 3, 6 ], [ 2, 4, 4 ], [ 6, 3, 1 ], [ 10, 0, 0 ] ]
gap> # this verifies that U is a distance-regular graph
gap> quit;
```

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