Polytope of Type {6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9}*324a
if this polytope has a name.
Group : SmallGroup(324,37)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 18, 81, 27
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,9,2} of size 648
   {6,9,4} of size 1296
   {6,9,6} of size 1944
Vertex Figure Of :
   {2,6,9} of size 648
   {3,6,9} of size 972
   {4,6,9} of size 1296
   {6,6,9} of size 1944
   {6,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,9}*108, {6,3}*108
   9-fold quotients : {2,9}*36, {6,3}*36
   27-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,18}*648a
   3-fold covers : {18,9}*972a, {6,27}*972a, {6,9}*972d, {18,9}*972h, {18,9}*972i, {6,9}*972e, {6,27}*972b, {6,27}*972c
   4-fold covers : {6,36}*1296a, {12,18}*1296e, {6,9}*1296b, {12,9}*1296c
   5-fold covers : {6,45}*1620a
   6-fold covers : {18,18}*1944b, {6,54}*1944a, {6,18}*1944h, {18,18}*1944w, {18,18}*1944aa, {6,18}*1944i, {6,54}*1944c, {6,54}*1944e, {6,18}*1944m
Permutation Representation (GAP) :

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

References : None.
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