Polytope of Type {2,18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,18}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1857)
Rank : 4
Schlafli Type : {2,18,18}
Number of vertices, edges, etc : 2, 18, 162, 18
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) :
   2-fold quotients : {2,9,18}*648
   3-fold quotients : {2,18,6}*432b
   6-fold quotients : {2,9,6}*216
   9-fold quotients : {2,18,2}*144, {2,6,6}*144c
   18-fold quotients : {2,9,2}*72, {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,  9)(  7, 11)(  8, 10)( 12, 23)( 13, 22)( 14, 21)( 15, 29)
( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 30, 63)( 31, 65)( 32, 64)
( 33, 60)( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 83)( 40, 82)
( 41, 81)( 42, 80)( 43, 79)( 44, 78)( 45, 77)( 46, 76)( 47, 75)( 48, 74)
( 49, 73)( 50, 72)( 51, 71)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 93,104)( 94,103)( 95,102)( 96,110)
( 97,109)( 98,108)( 99,107)(100,106)(101,105)(111,144)(112,146)(113,145)
(114,141)(115,143)(116,142)(117,138)(118,140)(119,139)(120,164)(121,163)
(122,162)(123,161)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)
(130,154)(131,153)(132,152)(133,151)(134,150)(135,149)(136,148)(137,147);;
s2 := (  3,120)(  4,122)(  5,121)(  6,126)(  7,128)(  8,127)(  9,123)( 10,125)
( 11,124)( 12,111)( 13,113)( 14,112)( 15,117)( 16,119)( 17,118)( 18,114)
( 19,116)( 20,115)( 21,131)( 22,130)( 23,129)( 24,137)( 25,136)( 26,135)
( 27,134)( 28,133)( 29,132)( 30, 93)( 31, 95)( 32, 94)( 33, 99)( 34,101)
( 35,100)( 36, 96)( 37, 98)( 38, 97)( 39, 84)( 40, 86)( 41, 85)( 42, 90)
( 43, 92)( 44, 91)( 45, 87)( 46, 89)( 47, 88)( 48,104)( 49,103)( 50,102)
( 51,110)( 52,109)( 53,108)( 54,107)( 55,106)( 56,105)( 57,153)( 58,155)
( 59,154)( 60,150)( 61,152)( 62,151)( 63,147)( 64,149)( 65,148)( 66,144)
( 67,146)( 68,145)( 69,141)( 70,143)( 71,142)( 72,138)( 73,140)( 74,139)
( 75,164)( 76,163)( 77,162)( 78,161)( 79,160)( 80,159)( 81,158)( 82,157)
( 83,156);;
s3 := (  4,  5)(  7,  8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)( 16, 25)
( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 34, 35)( 37, 38)( 39, 50)
( 40, 49)( 41, 48)( 42, 53)( 43, 52)( 44, 51)( 45, 56)( 46, 55)( 47, 54)
( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)( 68, 75)( 69, 80)( 70, 79)
( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 85, 86)( 88, 89)( 91, 92)( 93,104)
( 94,103)( 95,102)( 96,107)( 97,106)( 98,105)( 99,110)(100,109)(101,108)
(112,113)(115,116)(118,119)(120,131)(121,130)(122,129)(123,134)(124,133)
(125,132)(126,137)(127,136)(128,135)(139,140)(142,143)(145,146)(147,158)
(148,157)(149,156)(150,161)(151,160)(152,159)(153,164)(154,163)(155,162);;
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*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*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope