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Dear GAP-Forum,

Dear GAP Forum,

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

Thanks in advance,

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");

Loading GRAPE 2.31 (GRaph Algorithms using PErmutation groups),

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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