Polytope of Type {6,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*648b
if this polytope has a name.
Group : SmallGroup(648,548)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 9, 162, 54
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,36,2} of size 1296
Vertex Figure Of :
   {2,6,36} of size 1296
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,12}*216c
   9-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,36}*1296n
   3-fold covers : {6,36}*1944
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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