Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*648a
if this polytope has a name.
Group : SmallGroup(648,546)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 54, 162, 9
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {36,6,2} of size 1296
Vertex Figure Of :
   {2,36,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {12,6}*216c
   9-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,6}*1296m
   3-fold covers : {36,6}*1944, {108,6}*1944a, {108,6}*1944b, {108,6}*1944c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope