Polytope of Type {18,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,2}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1860)
Rank : 4
Schlafli Type : {18,6,2}
Number of vertices, edges, etc : 54, 162, 18, 2
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 : {9,6,2}*648c
   3-fold quotients : {6,6,2}*432c
   6-fold quotients : {3,6,2}*216
   9-fold quotients : {6,6,2}*144c
   18-fold quotients : {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 := (  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 60)
( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)( 39, 74)
( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)( 47, 66)
( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)( 83, 84)
( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)
( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,141)(113,140)
(114,139)(115,143)(116,142)(117,144)(118,154)(119,156)(120,155)(121,159)
(122,158)(123,157)(124,161)(125,160)(126,162)(127,145)(128,147)(129,146)
(130,150)(131,149)(132,148)(133,152)(134,151)(135,153);;
s1 := (  1,148)(  2,150)(  3,149)(  4,151)(  5,153)(  6,152)(  7,145)(  8,147)
(  9,146)( 10,142)( 11,144)( 12,143)( 13,136)( 14,138)( 15,137)( 16,139)
( 17,141)( 18,140)( 19,156)( 20,155)( 21,154)( 22,159)( 23,158)( 24,157)
( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,124)( 32,126)
( 33,125)( 34,118)( 35,120)( 36,119)( 37,115)( 38,117)( 39,116)( 40,109)
( 41,111)( 42,110)( 43,112)( 44,114)( 45,113)( 46,129)( 47,128)( 48,127)
( 49,132)( 50,131)( 51,130)( 52,135)( 53,134)( 54,133)( 55, 94)( 56, 96)
( 57, 95)( 58, 97)( 59, 99)( 60, 98)( 61, 91)( 62, 93)( 63, 92)( 64, 88)
( 65, 90)( 66, 89)( 67, 82)( 68, 84)( 69, 83)( 70, 85)( 71, 87)( 72, 86)
( 73,102)( 74,101)( 75,100)( 76,105)( 77,104)( 78,103)( 79,108)( 80,107)
( 81,106);;
s2 := (  4,  8)(  5,  9)(  6,  7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)( 91,100)
( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)( 99,104)
(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)(122,135)
(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)(145,154)
(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)(153,158);;
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope