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Polytope of Type {6,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*648a
if this polytope has a name.
Group : SmallGroup(648,546)
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}*1296m
   3-fold covers : {6,36}*1944, {6,108}*1944a, {6,108}*1944b, {6,108}*1944c
Permutation Representation (GAP) :
s0 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,55)
(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,73)(38,74)(39,75)
(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,64)(47,65)(48,66)(49,67)(50,68)
(51,69)(52,70)(53,71)(54,72);;
s1 := ( 1,28)( 2,30)( 3,29)( 4,36)( 5,35)( 6,34)( 7,33)( 8,32)( 9,31)(11,12)
(13,18)(14,17)(15,16)(19,73)(20,75)(21,74)(22,81)(23,80)(24,79)(25,78)(26,77)
(27,76)(37,64)(38,66)(39,65)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(47,48)
(49,54)(50,53)(51,52)(56,57)(58,63)(59,62)(60,61);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 7, 9)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)
(16,27)(17,26)(18,25)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,45)(35,44)
(36,43)(46,49)(47,51)(48,50)(52,54)(55,76)(56,78)(57,77)(58,73)(59,75)(60,74)
(61,81)(62,80)(63,79)(64,67)(65,69)(66,68)(70,72);;
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*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(81)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)
(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,73)(38,74)
(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,64)(47,65)(48,66)(49,67)
(50,68)(51,69)(52,70)(53,71)(54,72);
s1 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,36)( 5,35)( 6,34)( 7,33)( 8,32)( 9,31)
(11,12)(13,18)(14,17)(15,16)(19,73)(20,75)(21,74)(22,81)(23,80)(24,79)(25,78)
(26,77)(27,76)(37,64)(38,66)(39,65)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)
(47,48)(49,54)(50,53)(51,52)(56,57)(58,63)(59,62)(60,61);
s2 := Sym(81)!( 1, 4)( 2, 6)( 3, 5)( 7, 9)(10,22)(11,24)(12,23)(13,19)(14,21)
(15,20)(16,27)(17,26)(18,25)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,45)
(35,44)(36,43)(46,49)(47,51)(48,50)(52,54)(55,76)(56,78)(57,77)(58,73)(59,75)
(60,74)(61,81)(62,80)(63,79)(64,67)(65,69)(66,68)(70,72);
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*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
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