Polytope of Type {6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*324a
if this polytope has a name.
Group : SmallGroup(324,37)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 9, 81, 27
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,18,2} of size 648
   {6,18,4} of size 1296
   {6,18,6} of size 1944
Vertex Figure Of :
   {2,6,18} of size 648
   {4,6,18} of size 1296
   {6,6,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,18}*648b
   3-fold covers : {18,18}*972a, {6,54}*972a, {6,18}*972c, {18,18}*972g, {18,18}*972i, {6,18}*972d, {6,54}*972b, {6,54}*972c
   4-fold covers : {12,18}*1296a, {6,36}*1296b, {6,36}*1296k, {12,18}*1296k
   5-fold covers : {6,90}*1620a, {30,18}*1620a
   6-fold covers : {18,18}*1944c, {6,54}*1944b, {6,18}*1944g, {18,18}*1944v, {18,18}*1944z, {6,18}*1944j, {6,54}*1944d, {6,54}*1944f, {6,18}*1944n
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)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,38)
(11,37)(12,39)(13,44)(14,43)(15,45)(16,41)(17,40)(18,42)(19,48)(20,47)(21,46)
(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(55,58)(56,60)(57,59)(62,63)(64,68)
(65,67)(66,69)(70,71)(73,78)(74,77)(75,76)(79,81);;
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*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,38)(11,37)(12,39)(13,44)(14,43)(15,45)(16,41)(17,40)(18,42)(19,48)(20,47)
(21,46)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(55,58)(56,60)(57,59)(62,63)
(64,68)(65,67)(66,69)(70,71)(73,78)(74,77)(75,76)(79,81);
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*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2 >;

References : None.
to this polytope