Polytope of Type {2,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,18}*1296h
if this polytope has a name.
Group : SmallGroup(1296,1862)
Rank : 4
Schlafli Type : {2,6,18}
Number of vertices, edges, etc : 2, 18, 162, 54
Order of s₀s₁s₂s₃ : 6
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) :
   2-fold quotients : {2,3,18}*648
   3-fold quotients : {2,6,6}*432c
   6-fold quotients : {2,3,6}*216
   9-fold quotients : {2,6,6}*144c
   18-fold quotients : {2,3,6}*72
   27-fold quotients : {2,6,2}*48
   54-fold quotients : {2,3,2}*24
   81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  6,  8)(  9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)( 16, 25)
( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 59)( 31, 58)( 32, 57)( 33, 61)
( 34, 60)( 35, 62)( 36, 63)( 37, 65)( 38, 64)( 39, 77)( 40, 76)( 41, 75)
( 42, 79)( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 68)( 49, 67)
( 50, 66)( 51, 70)( 52, 69)( 53, 71)( 54, 72)( 55, 74)( 56, 73)( 85, 86)
( 87, 89)( 90, 91)( 93,102)( 94,104)( 95,103)( 96,107)( 97,106)( 98,105)
( 99,109)(100,108)(101,110)(111,140)(112,139)(113,138)(114,142)(115,141)
(116,143)(117,144)(118,146)(119,145)(120,158)(121,157)(122,156)(123,160)
(124,159)(125,161)(126,162)(127,164)(128,163)(129,149)(130,148)(131,147)
(132,151)(133,150)(134,152)(135,153)(136,155)(137,154);;
s2 := (  3,150)(  4,152)(  5,151)(  6,153)(  7,155)(  8,154)(  9,147)( 10,149)
( 11,148)( 12,144)( 13,146)( 14,145)( 15,138)( 16,140)( 17,139)( 18,141)
( 19,143)( 20,142)( 21,158)( 22,157)( 23,156)( 24,161)( 25,160)( 26,159)
( 27,164)( 28,163)( 29,162)( 30,123)( 31,125)( 32,124)( 33,126)( 34,128)
( 35,127)( 36,120)( 37,122)( 38,121)( 39,117)( 40,119)( 41,118)( 42,111)
( 43,113)( 44,112)( 45,114)( 46,116)( 47,115)( 48,131)( 49,130)( 50,129)
( 51,134)( 52,133)( 53,132)( 54,137)( 55,136)( 56,135)( 57, 96)( 58, 98)
( 59, 97)( 60, 99)( 61,101)( 62,100)( 63, 93)( 64, 95)( 65, 94)( 66, 90)
( 67, 92)( 68, 91)( 69, 84)( 70, 86)( 71, 85)( 72, 87)( 73, 89)( 74, 88)
( 75,104)( 76,103)( 77,102)( 78,107)( 79,106)( 80,105)( 81,110)( 82,109)
( 83,108);;
s3 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 59)( 31, 58)( 32, 57)( 33, 65)
( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 68)( 40, 67)( 41, 66)
( 42, 74)( 43, 73)( 44, 72)( 45, 71)( 46, 70)( 47, 69)( 48, 77)( 49, 76)
( 50, 75)( 51, 83)( 52, 82)( 53, 81)( 54, 80)( 55, 79)( 56, 78)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,140)(112,139)(113,138)(114,146)(115,145)
(116,144)(117,143)(118,142)(119,141)(120,149)(121,148)(122,147)(123,155)
(124,154)(125,153)(126,152)(127,151)(128,150)(129,158)(130,157)(131,156)
(132,164)(133,163)(134,162)(135,161)(136,160)(137,159);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(164)!(1,2);
s1 := Sym(164)!(  4,  5)(  6,  8)(  9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)
( 16, 25)( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 59)( 31, 58)( 32, 57)
( 33, 61)( 34, 60)( 35, 62)( 36, 63)( 37, 65)( 38, 64)( 39, 77)( 40, 76)
( 41, 75)( 42, 79)( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 68)
( 49, 67)( 50, 66)( 51, 70)( 52, 69)( 53, 71)( 54, 72)( 55, 74)( 56, 73)
( 85, 86)( 87, 89)( 90, 91)( 93,102)( 94,104)( 95,103)( 96,107)( 97,106)
( 98,105)( 99,109)(100,108)(101,110)(111,140)(112,139)(113,138)(114,142)
(115,141)(116,143)(117,144)(118,146)(119,145)(120,158)(121,157)(122,156)
(123,160)(124,159)(125,161)(126,162)(127,164)(128,163)(129,149)(130,148)
(131,147)(132,151)(133,150)(134,152)(135,153)(136,155)(137,154);
s2 := Sym(164)!(  3,150)(  4,152)(  5,151)(  6,153)(  7,155)(  8,154)(  9,147)
( 10,149)( 11,148)( 12,144)( 13,146)( 14,145)( 15,138)( 16,140)( 17,139)
( 18,141)( 19,143)( 20,142)( 21,158)( 22,157)( 23,156)( 24,161)( 25,160)
( 26,159)( 27,164)( 28,163)( 29,162)( 30,123)( 31,125)( 32,124)( 33,126)
( 34,128)( 35,127)( 36,120)( 37,122)( 38,121)( 39,117)( 40,119)( 41,118)
( 42,111)( 43,113)( 44,112)( 45,114)( 46,116)( 47,115)( 48,131)( 49,130)
( 50,129)( 51,134)( 52,133)( 53,132)( 54,137)( 55,136)( 56,135)( 57, 96)
( 58, 98)( 59, 97)( 60, 99)( 61,101)( 62,100)( 63, 93)( 64, 95)( 65, 94)
( 66, 90)( 67, 92)( 68, 91)( 69, 84)( 70, 86)( 71, 85)( 72, 87)( 73, 89)
( 74, 88)( 75,104)( 76,103)( 77,102)( 78,107)( 79,106)( 80,105)( 81,110)
( 82,109)( 83,108);
s3 := Sym(164)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 59)( 31, 58)( 32, 57)
( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 68)( 40, 67)
( 41, 66)( 42, 74)( 43, 73)( 44, 72)( 45, 71)( 46, 70)( 47, 69)( 48, 77)
( 49, 76)( 50, 75)( 51, 83)( 52, 82)( 53, 81)( 54, 80)( 55, 79)( 56, 78)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,140)(112,139)(113,138)(114,146)
(115,145)(116,144)(117,143)(118,142)(119,141)(120,149)(121,148)(122,147)
(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,158)(130,157)
(131,156)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159);
poly := sub<Sym(164)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 >;

to this polytope