Polytope of Type {18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*324b
if this polytope has a name.
Group : SmallGroup(324,40)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 27, 81, 9
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {18,6,2} of size 648
   {18,6,4} of size 1296
   {18,6,6} of size 1944
Vertex Figure Of :
   {2,18,6} of size 648
   {4,18,6} of size 1296
   {6,18,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,6}*648f
   3-fold covers : {18,6}*972a, {18,18}*972c, {18,18}*972e, {18,18}*972g, {18,6}*972d
   4-fold covers : {18,12}*1296b, {36,6}*1296f, {36,6}*1296i, {18,12}*1296j
   5-fold covers : {90,6}*1620b, {18,30}*1620b
   6-fold covers : {18,6}*1944c, {18,18}*1944k, {18,18}*1944m, {18,18}*1944v, {18,6}*1944j, {18,6}*1944s
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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