Polytope of Type {3,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,3}*972
if this polytope has a name.
Group : SmallGroup(972,101)
Rank : 4
Schlafli Type : {3,6,3}
Number of vertices, edges, etc : 9, 81, 81, 9
Order of s₀s₁s₂s₃ : 9
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Locally Toroidal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,3,2} of size 1944
Vertex Figure Of :
   {2,3,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6,3}*324a, {3,6,3}*324b
   9-fold quotients : {3,6,3}*108
   27-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,6}*1944a, {6,6,3}*1944a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope