Polytope of Type {6,3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,6}*1944d
Also Known As : ₆T⁴_(1,1)(3,3). if this polytope has another name.
Group : SmallGroup(1944,2346)
Rank : 4
Schlafli Type : {6,3,6}
Number of vertices, edges, etc : 6, 81, 81, 54
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Locally Toroidal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,3,6}*648, {6,3,6}*648a
   9-fold quotients : {2,3,6}*216, {6,3,6}*216
   27-fold quotients : {2,3,6}*72, {6,3,2}*72
   81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)
(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)
(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);;
s1 := ( 1,28)( 2,30)( 3,29)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)(10,47)
(11,46)(12,48)(13,50)(14,49)(15,51)(16,53)(17,52)(18,54)(19,38)(20,37)(21,39)
(22,41)(23,40)(24,42)(25,44)(26,43)(27,45)(56,57)(59,60)(62,63)(64,74)(65,73)
(66,75)(67,77)(68,76)(69,78)(70,80)(71,79)(72,81);;
s2 := ( 1,13)( 2,15)( 3,14)( 4,18)( 5,17)( 6,16)( 7,11)( 8,10)( 9,12)(20,21)
(22,24)(25,26)(28,67)(29,69)(30,68)(31,72)(32,71)(33,70)(34,65)(35,64)(36,66)
(37,62)(38,61)(39,63)(40,55)(41,57)(42,56)(43,60)(44,59)(45,58)(46,73)(47,75)
(48,74)(49,78)(50,77)(51,76)(52,80)(53,79)(54,81);;
s3 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)
(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)
(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)
(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);
s1 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)
(10,47)(11,46)(12,48)(13,50)(14,49)(15,51)(16,53)(17,52)(18,54)(19,38)(20,37)
(21,39)(22,41)(23,40)(24,42)(25,44)(26,43)(27,45)(56,57)(59,60)(62,63)(64,74)
(65,73)(66,75)(67,77)(68,76)(69,78)(70,80)(71,79)(72,81);
s2 := Sym(81)!( 1,13)( 2,15)( 3,14)( 4,18)( 5,17)( 6,16)( 7,11)( 8,10)( 9,12)
(20,21)(22,24)(25,26)(28,67)(29,69)(30,68)(31,72)(32,71)(33,70)(34,65)(35,64)
(36,66)(37,62)(38,61)(39,63)(40,55)(41,57)(42,56)(43,60)(44,59)(45,58)(46,73)
(47,75)(48,74)(49,78)(50,77)(51,76)(52,80)(53,79)(54,81);
s3 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)
(60,63)(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);
poly := sub<Sym(81)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 >;

References :

Theorem 11C7,11C8, McMullen P., Schulte, E.; Abstract Regular Polytopes (\ Cambridge University Press, 2002)

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