# SageMathCell online
# https://sagecell.sagemath.org/?q=iwzbqb

#An utility to deal with the vertex set of edges, without using python sets.
def sorttuple(t):
    if t[1]<t[0]:
        return((t[1],t[0]))
    return(t)

#The variable Coxt is the Coxeter graph, with 28 vtx, in graph6 format .
#Coxtr=Graph("[hCGIC@?G?_@?@_?_?G?`@?C??G??G??C??@?G?G???o??`?__OA@?AA?CC@?OC?")

#The variable Coxt is the Coxeter graph, with 28 vtx, in sparse6 format .
Coxtr=Graph(":[_`abcd`Efghij_KlhMdNopqrgSt_PUeKRcJUbMSaIQ")

print("Coxeter. Order: ",Coxtr.order(),"Is 3-regular? ",Coxtr.is_regular(k=3)," Girth: ", Coxtr.girth()," Diameter: ",Coxtr.diameter())

#AutCoxtr is the automorphism group of Coxtr.
AutCoxtr=Coxtr.automorphism_group()

#Extract the index-2 normal subgroup of AutCoxtr
ConjSubgroups=AutCoxtr.conjugacy_classes_subgroups()
Index2=ConjSubgroups[-2]

# Pairing is the edge pairing to construct the diameter-5 graph
# pairings that produce the graph  (0,1),(19,20) -- (0,13),(4,5)   -- (0,23),(9,10) 
Pairing = Index2.orbit(((0,1),(19,20)), action = "OnTuplesTuples")

#G will be a diameter-5 3-regular graph on 70 vertices.

G=Graph()
for q in Pairing:
    x=q[0]
    y=q[1]

    #If v?E(G), u?E(G), then they are connected in H iff v~u by the pairing relation.
    G.add_edge(sorttuple(x),sorttuple(y))
   
    G.add_edge(sorttuple(x),x[0])
    G.add_edge(sorttuple(x),x[1])
    G.add_edge(sorttuple(y),y[0])
    G.add_edge(sorttuple(y),y[1])
    
#G is a diameter-5 3-regular graph on 70 vertices.
print("\nFC graph isomorhic to vC, AFY. Order: ",G.order()," Size:",G.size(), "Is 3-regular? ",G.is_regular(k=3)," Girth: ", G.girth()," Diameter: ",G.diameter())

#print("\n in graph6 format")
#print(">>graph6<<"+G.graph6_string())

#print("\n in sparse6 format")
print(">>sparse6<<"+G.sparse6_string())

# >>sparse6<<:~?@E_GA?_WEF@gYE_giJaw{Mbg}GchMAdXUUdguXEhi[fXu]fh?VfAIbaAQPHYYHEWS`IHA_Iiij_p}HJAyBCYso`Wsqd@}slYAuLwok`BWxdyCfMxOWbA[|`AQWJR}ICb]FKRMFLRmMCrINErb