Polytope of Type {18,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,2}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 4
Schlafli Type : {18,6,2}
Number of vertices, edges, etc : 54, 162, 18, 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) :
   2-fold quotients : {9,6,2}*648a
   3-fold quotients : {18,6,2}*432b, {6,6,2}*432c
   6-fold quotients : {9,6,2}*216, {3,6,2}*216
   9-fold quotients : {18,2,2}*144, {6,6,2}*144c
   18-fold quotients : {9,2,2}*72, {3,6,2}*72
   27-fold quotients : {6,2,2}*48
   54-fold quotients : {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)( 28, 61)( 29, 62)( 30, 63)( 31, 58)
( 32, 59)( 33, 60)( 34, 55)( 35, 56)( 36, 57)( 37, 79)( 38, 80)( 39, 81)
( 40, 76)( 41, 77)( 42, 78)( 43, 73)( 44, 74)( 45, 75)( 46, 70)( 47, 71)
( 48, 72)( 49, 67)( 50, 68)( 51, 69)( 52, 64)( 53, 65)( 54, 66)( 85, 88)
( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)
( 97,103)( 98,104)( 99,105)(109,142)(110,143)(111,144)(112,139)(113,140)
(114,141)(115,136)(116,137)(117,138)(118,160)(119,161)(120,162)(121,157)
(122,158)(123,159)(124,154)(125,155)(126,156)(127,151)(128,152)(129,153)
(130,148)(131,149)(132,150)(133,145)(134,146)(135,147);;
s1 := (  1,146)(  2,147)(  3,145)(  4,152)(  5,153)(  6,151)(  7,149)(  8,150)
(  9,148)( 10,138)( 11,136)( 12,137)( 13,144)( 14,142)( 15,143)( 16,141)
( 17,139)( 18,140)( 19,154)( 20,155)( 21,156)( 22,160)( 23,161)( 24,162)
( 25,157)( 26,158)( 27,159)( 28,119)( 29,120)( 30,118)( 31,125)( 32,126)
( 33,124)( 34,122)( 35,123)( 36,121)( 37,111)( 38,109)( 39,110)( 40,117)
( 41,115)( 42,116)( 43,114)( 44,112)( 45,113)( 46,127)( 47,128)( 48,129)
( 49,133)( 50,134)( 51,135)( 52,130)( 53,131)( 54,132)( 55, 92)( 56, 93)
( 57, 91)( 58, 98)( 59, 99)( 60, 97)( 61, 95)( 62, 96)( 63, 94)( 64, 84)
( 65, 82)( 66, 83)( 67, 90)( 68, 88)( 69, 89)( 70, 87)( 71, 85)( 72, 86)
( 73,100)( 74,101)( 75,102)( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)
( 81,105);;
s2 := (  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)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161);;
s3 := (163,164);;
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope