Polytope of Type {3,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,18}*1944e
Also Known As : {{3,6}₆,{6,18|2}}. if this polytope has another name.
Group : SmallGroup(1944,2341)
Rank : 4
Schlafli Type : {3,6,18}
Number of vertices, edges, etc : 9, 27, 162, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   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 : {3,6,18}*648b, {3,6,6}*648c
   9-fold quotients : {3,2,18}*216, {3,6,2}*216, {3,6,6}*216b
   18-fold quotients : {3,2,9}*108
   27-fold quotients : {3,2,6}*72, {3,6,2}*72
   54-fold quotients : {3,2,3}*36
   81-fold quotients : {3,2,2}*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 := ( 1,29)( 2,30)( 3,28)( 4,32)( 5,33)( 6,31)( 7,35)( 8,36)( 9,34)(10,38)
(11,39)(12,37)(13,41)(14,42)(15,40)(16,44)(17,45)(18,43)(19,47)(20,48)(21,46)
(22,50)(23,51)(24,49)(25,53)(26,54)(27,52);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)
(16,25)(17,27)(18,26)(29,30)(31,34)(32,36)(33,35)(37,49)(38,51)(39,50)(40,46)
(41,48)(42,47)(43,52)(44,54)(45,53)(56,57)(58,61)(59,63)(60,62)(64,76)(65,78)
(66,77)(67,73)(68,75)(69,74)(70,79)(71,81)(72,80);;
s3 := ( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)(19,22)
(20,23)(21,24)(28,37)(29,38)(30,39)(31,43)(32,44)(33,45)(34,40)(35,41)(36,42)
(46,49)(47,50)(48,51)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)
(63,69)(73,76)(74,77)(75,78);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, 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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)!( 1,29)( 2,30)( 3,28)( 4,32)( 5,33)( 6,31)( 7,35)( 8,36)( 9,34)
(10,38)(11,39)(12,37)(13,41)(14,42)(15,40)(16,44)(17,45)(18,43)(19,47)(20,48)
(21,46)(22,50)(23,51)(24,49)(25,53)(26,54)(27,52);
s2 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,22)(11,24)(12,23)(13,19)(14,21)
(15,20)(16,25)(17,27)(18,26)(29,30)(31,34)(32,36)(33,35)(37,49)(38,51)(39,50)
(40,46)(41,48)(42,47)(43,52)(44,54)(45,53)(56,57)(58,61)(59,63)(60,62)(64,76)
(65,78)(66,77)(67,73)(68,75)(69,74)(70,79)(71,81)(72,80);
s3 := Sym(81)!( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)
(19,22)(20,23)(21,24)(28,37)(29,38)(30,39)(31,43)(32,44)(33,45)(34,40)(35,41)
(36,42)(46,49)(47,50)(48,51)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)
(62,68)(63,69)(73,76)(74,77)(75,78);
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, s1*s2*s3*s2*s1*s2*s3*s2, 
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope