#   https://sagecell.sagemath.org/?q=cacfyl

#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 F144A is the F144A graph in sparse6 format.
F144A=Graph(r":~?AO_C?_CF?_D?sC_sD`CI`KQ@sO`{E_cD`kJ`c]ACM`{L`[Kcg]Cc^Cc_CcVCSUCKTC[QCKSC[RCSWB[YBkXBcPaKPaSSa[Ta{UbCXbS^cC]b[\bc[bkZh@CG|DG|BG{~Gk|Gc}G[tFSsF[uFc{F{yFszFkrE{qFKpFCpGCrGSqGKjGslGskGsgGCiGShGKmEcoEsnEkdE{fFKeFCNaCMjXejPfj`gi`fipgihehpVi@XhxWiXaiP`iH_hhdh`bhXch@\hP^hH]`CI`LvN\uNltNdqN\sNdrNlkL|mMLlMDhNDjNTiNKANyF_I?PKBNqG_I@OcBOYE_QAOn")

#AutF144A is the automorphism group of F144A.
AutF144A=F144A.automorphism_group()
print(F144A.edges())

#Extract the index-2 normal subgroup of AutF144A
ConjSubgroups=AutF144A.conjugacy_classes_subgroups()
Index2=ConjSubgroups[-2]

#Pairing is the edge pairing to construct the diameter-8 graph
Pairing = Index2.orbit(((0,1),(28,66)), action = "OnTuplesTuples")

#G will be a diameter-8 3-regular graph on 360 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))
    
    #If v?V(G), u?E(G), then they are connected in H iff v?u in G.
    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-8 3-regular graph on 360 vertices 540 edges with girth 13.
print(G.order(),G.size(),G.is_regular(k=3),G.girth(),G.diameter())

# Next output is G in sparse6 format.  
print(">>sparse6<<"+G.sparse6_string())

#  >>sparse6<<:~?Dg_A?G?_E?WF?Y@gE_]B?J_mAwK_qC_P`ECWQ`IE?V`]DwW`aF_\`uFW]`y?gRAMGwca_HwfaaJ?jamJGkbCKGobAKWuBULWtbYMoybiMgzbmDgnCAOH@cSPHCcQPXDckQhIciQxJdCSHPdETPSdUTWGccUHWdeVP[dqVH\eGWX`eIX_LEUXXeegYXheeYhif?Zxne}[HtCK\HsfQ\Wmac]Xxfi^`|fu^X}fy`@zGM_yBgQEX]EKaG^DIMAI_G[xkGqbqMgybiNg}dODHQdIThceIWhaeYXcW`xKHyfyaIEgY`iIggOGY\qfigiYhieiiiaUUp{I}jwrJERaqjMlaUJUlYtjYGHmJimy}BGnY|jyo`kHsoZ@_sjXKHCpJFKKpjEkYpzKKmqzJkqrI{lEsZQlImHNLYty~aWuZXliQBZasvZ]e[vzWMAwWvBuTbIMMN@^l?xWUD}KakM]oHYMeyZi_wlwSMozXgh[lBnm}cq[g?{iAIq[rs_SWGGHG|gCMILBwneOi_Ng}zzcs\gBMk~hlIYTqPOAFq_gGdcAekrg[GD@HpLQSqcOYNPzholsGf}WAbOiDQD_g{SK`cJWrHLBgRKeEpEPAEqYaG[cQiSqG`IxDIQKuqRRPYSgYPaM@HetEgNNuAR[Pq`jRL`Fjkk{}c_eOcG}QIowOQQPwCIxHgiJePpwQaMRNhkisi`gNylLTJGnFIRsmkA`qbRAGApdW]Cq_yQCsaGUg{E\Lg`GUrbZRaBo[_KLsyaW^w~F\NIeJaHRORyL@p_gqd?eaOrLSIXBafo_TCfWeycL|PhMEmD@RSaJ`m_citIcsVghK`RIxLQWr[SyDQOagMTOdktXAHtSi{Mm_BpTQdqYd[ntUbwlIdNtUGFLXEWjHPHGfJtEGgESbWpI`EwuOhJx\JhLH`GlOhaKDKHONLHhOMXHXSG`GHoQLLXSNTFn