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# Polytope of Type {6,18}

Atlas Canonical Name : {6,18}*324b
if this polytope has a name.
Group : SmallGroup(324,40)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 9, 81, 27
Order of s0s1s2 : 9
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,18,2} of size 648
{6,18,4} of size 1296
{6,18,6} of size 1944
Vertex Figure Of :
{2,6,18} of size 648
{4,6,18} of size 1296
{6,6,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,18}*648f
3-fold covers : {6,18}*972a, {18,18}*972d, {18,18}*972e, {18,18}*972f, {6,18}*972d
4-fold covers : {12,18}*1296b, {6,36}*1296f, {6,36}*1296i, {12,18}*1296j
5-fold covers : {6,90}*1620b, {30,18}*1620b
6-fold covers : {6,18}*1944c, {18,18}*1944i, {18,18}*1944m, {18,18}*1944s, {6,18}*1944j, {6,18}*1944s
Permutation Representation (GAP) :
s0 := ( 4, 8)( 5, 9)( 6, 7)(10,19)(11,20)(12,21)(13,26)(14,27)(15,25)(16,24)
(17,22)(18,23)(31,35)(32,36)(33,34)(37,46)(38,47)(39,48)(40,53)(41,54)(42,52)
(43,51)(44,49)(45,50)(58,62)(59,63)(60,61)(64,73)(65,74)(66,75)(67,80)(68,81)
(69,79)(70,78)(71,76)(72,77);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)
(23,24)(26,27)(28,65)(29,64)(30,66)(31,68)(32,67)(33,69)(34,71)(35,70)(36,72)
(37,56)(38,55)(39,57)(40,59)(41,58)(42,60)(43,62)(44,61)(45,63)(46,74)(47,73)
(48,75)(49,77)(50,76)(51,78)(52,80)(53,79)(54,81);;
s2 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,40)
(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,53)(20,52)(21,54)
(22,50)(23,49)(24,51)(25,47)(26,46)(27,48)(55,56)(58,62)(59,61)(60,63)(64,68)
(65,67)(66,69)(70,71)(73,81)(74,80)(75,79)(76,78);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(81)!( 4, 8)( 5, 9)( 6, 7)(10,19)(11,20)(12,21)(13,26)(14,27)(15,25)
(16,24)(17,22)(18,23)(31,35)(32,36)(33,34)(37,46)(38,47)(39,48)(40,53)(41,54)
(42,52)(43,51)(44,49)(45,50)(58,62)(59,63)(60,61)(64,73)(65,74)(66,75)(67,80)
(68,81)(69,79)(70,78)(71,76)(72,77);
s1 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)
(20,21)(23,24)(26,27)(28,65)(29,64)(30,66)(31,68)(32,67)(33,69)(34,71)(35,70)
(36,72)(37,56)(38,55)(39,57)(40,59)(41,58)(42,60)(43,62)(44,61)(45,63)(46,74)
(47,73)(48,75)(49,77)(50,76)(51,78)(52,80)(53,79)(54,81);
s2 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,40)(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,53)(20,52)
(21,54)(22,50)(23,49)(24,51)(25,47)(26,46)(27,48)(55,56)(58,62)(59,61)(60,63)
(64,68)(65,67)(66,69)(70,71)(73,81)(74,80)(75,79)(76,78);
poly := sub<Sym(81)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0 >;

References : None.
to this polytope