Polytope of Type {6,18,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18,2}*1296i
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 4
Schlafli Type : {6,18,2}
Number of vertices, edges, etc : 18, 162, 54, 2
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {6,18,2}*432a, {6,18,2}*432b, {6,6,2}*432d
   6-fold quotients : {6,9,2}*216
   9-fold quotients : {2,18,2}*144, {6,6,2}*144a, {6,6,2}*144b, {6,6,2}*144c
   18-fold quotients : {2,9,2}*72, {3,6,2}*72, {6,3,2}*72
   27-fold quotients : {2,6,2}*48, {6,2,2}*48
   54-fold quotients : {2,3,2}*24, {3,2,2}*24
   81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)
(17,23)(18,24)(31,34)(32,35)(33,36)(37,46)(38,47)(39,48)(40,52)(41,53)(42,54)
(43,49)(44,50)(45,51)(58,61)(59,62)(60,63)(64,73)(65,74)(66,75)(67,79)(68,80)
(69,81)(70,76)(71,77)(72,78);;
s1 := ( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)
(20,24)(21,23)(26,27)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,72)(35,71)
(36,70)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,63)(44,62)(45,61)(46,78)
(47,77)(48,76)(49,75)(50,74)(51,73)(52,81)(53,80)(54,79);;
s2 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,37)
(11,39)(12,38)(13,43)(14,45)(15,44)(16,40)(17,42)(18,41)(19,46)(20,48)(21,47)
(22,52)(23,54)(24,53)(25,49)(26,51)(27,50)(55,57)(58,63)(59,62)(60,61)(64,66)
(67,72)(68,71)(69,70)(73,75)(76,81)(77,80)(78,79);;
s3 := (82,83);;
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, s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(83)!( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)
(16,22)(17,23)(18,24)(31,34)(32,35)(33,36)(37,46)(38,47)(39,48)(40,52)(41,53)
(42,54)(43,49)(44,50)(45,51)(58,61)(59,62)(60,63)(64,73)(65,74)(66,75)(67,79)
(68,80)(69,81)(70,76)(71,77)(72,78);
s1 := Sym(83)!( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)
(19,22)(20,24)(21,23)(26,27)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,72)
(35,71)(36,70)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,63)(44,62)(45,61)
(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,81)(53,80)(54,79);
s2 := Sym(83)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,37)(11,39)(12,38)(13,43)(14,45)(15,44)(16,40)(17,42)(18,41)(19,46)(20,48)
(21,47)(22,52)(23,54)(24,53)(25,49)(26,51)(27,50)(55,57)(58,63)(59,62)(60,61)
(64,66)(67,72)(68,71)(69,70)(73,75)(76,81)(77,80)(78,79);
s3 := Sym(83)!(82,83);
poly := sub<Sym(83)|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, 
s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope