Polytope of Type {2,18,6}

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

Permutation Representation (Magma) :

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

to this polytope