Polytope of Type {6,18,2}

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