Polytope of Type {18,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,6}*1296d
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 4
Schlafli Type : {18,6,6}
Number of vertices, edges, etc : 18, 54, 18, 6
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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) :
   3-fold quotients : {18,6,2}*432a, {6,6,6}*432e
   9-fold quotients : {18,2,2}*144, {2,6,6}*144c, {6,6,2}*144a
   18-fold quotients : {9,2,2}*72, {2,3,6}*72
   27-fold quotients : {2,6,2}*48, {6,2,2}*48
   54-fold quotients : {2,3,2}*24, {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)(  5,  6)(  8,  9)( 10, 20)( 11, 19)( 12, 21)( 13, 23)( 14, 22)
( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 29, 30)( 32, 33)( 35, 36)( 37, 47)
( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)
( 56, 57)( 59, 60)( 62, 63)( 64, 74)( 65, 73)( 66, 75)( 67, 77)( 68, 76)
( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 83, 84)( 86, 87)( 89, 90)( 91,101)
( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)
(110,111)(113,114)(116,117)(118,128)(119,127)(120,129)(121,131)(122,130)
(123,132)(124,134)(125,133)(126,135)(137,138)(140,141)(143,144)(145,155)
(146,154)(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162);;
s1 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 64)( 29, 66)( 30, 65)
( 31, 70)( 32, 72)( 33, 71)( 34, 67)( 35, 69)( 36, 68)( 37, 55)( 38, 57)
( 39, 56)( 40, 61)( 41, 63)( 42, 62)( 43, 58)( 44, 60)( 45, 59)( 46, 74)
( 47, 73)( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,101)(103,107)(104,106)(105,108)(109,145)(110,147)(111,146)
(112,151)(113,153)(114,152)(115,148)(116,150)(117,149)(118,136)(119,138)
(120,137)(121,142)(122,144)(123,143)(124,139)(125,141)(126,140)(127,155)
(128,154)(129,156)(130,161)(131,160)(132,162)(133,158)(134,157)(135,159);;
s2 := (  1,112)(  2,113)(  3,114)(  4,109)(  5,110)(  6,111)(  7,115)(  8,116)
(  9,117)( 10,121)( 11,122)( 12,123)( 13,118)( 14,119)( 15,120)( 16,124)
( 17,125)( 18,126)( 19,130)( 20,131)( 21,132)( 22,127)( 23,128)( 24,129)
( 25,133)( 26,134)( 27,135)( 28, 85)( 29, 86)( 30, 87)( 31, 82)( 32, 83)
( 33, 84)( 34, 88)( 35, 89)( 36, 90)( 37, 94)( 38, 95)( 39, 96)( 40, 91)
( 41, 92)( 42, 93)( 43, 97)( 44, 98)( 45, 99)( 46,103)( 47,104)( 48,105)
( 49,100)( 50,101)( 51,102)( 52,106)( 53,107)( 54,108)( 55,139)( 56,140)
( 57,141)( 58,136)( 59,137)( 60,138)( 61,142)( 62,143)( 63,144)( 64,148)
( 65,149)( 66,150)( 67,145)( 68,146)( 69,147)( 70,151)( 71,152)( 72,153)
( 73,157)( 74,158)( 75,159)( 76,154)( 77,155)( 78,156)( 79,160)( 80,161)
( 81,162);;
s3 := ( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)
( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)
( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)
( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)(113,140)
(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)
(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)
(130,157)(131,158)(132,159)(133,160)(134,161)(135,162);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
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(162)!(  2,  3)(  5,  6)(  8,  9)( 10, 20)( 11, 19)( 12, 21)( 13, 23)
( 14, 22)( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 29, 30)( 32, 33)( 35, 36)
( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)
( 45, 54)( 56, 57)( 59, 60)( 62, 63)( 64, 74)( 65, 73)( 66, 75)( 67, 77)
( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 83, 84)( 86, 87)( 89, 90)
( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)
( 99,108)(110,111)(113,114)(116,117)(118,128)(119,127)(120,129)(121,131)
(122,130)(123,132)(124,134)(125,133)(126,135)(137,138)(140,141)(143,144)
(145,155)(146,154)(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)
(153,162);
s1 := Sym(162)!(  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)
(  8, 15)(  9, 14)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 64)( 29, 66)
( 30, 65)( 31, 70)( 32, 72)( 33, 71)( 34, 67)( 35, 69)( 36, 68)( 37, 55)
( 38, 57)( 39, 56)( 40, 61)( 41, 63)( 42, 62)( 43, 58)( 44, 60)( 45, 59)
( 46, 74)( 47, 73)( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)
( 54, 78)( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)
( 89, 96)( 90, 95)(100,101)(103,107)(104,106)(105,108)(109,145)(110,147)
(111,146)(112,151)(113,153)(114,152)(115,148)(116,150)(117,149)(118,136)
(119,138)(120,137)(121,142)(122,144)(123,143)(124,139)(125,141)(126,140)
(127,155)(128,154)(129,156)(130,161)(131,160)(132,162)(133,158)(134,157)
(135,159);
s2 := Sym(162)!(  1,112)(  2,113)(  3,114)(  4,109)(  5,110)(  6,111)(  7,115)
(  8,116)(  9,117)( 10,121)( 11,122)( 12,123)( 13,118)( 14,119)( 15,120)
( 16,124)( 17,125)( 18,126)( 19,130)( 20,131)( 21,132)( 22,127)( 23,128)
( 24,129)( 25,133)( 26,134)( 27,135)( 28, 85)( 29, 86)( 30, 87)( 31, 82)
( 32, 83)( 33, 84)( 34, 88)( 35, 89)( 36, 90)( 37, 94)( 38, 95)( 39, 96)
( 40, 91)( 41, 92)( 42, 93)( 43, 97)( 44, 98)( 45, 99)( 46,103)( 47,104)
( 48,105)( 49,100)( 50,101)( 51,102)( 52,106)( 53,107)( 54,108)( 55,139)
( 56,140)( 57,141)( 58,136)( 59,137)( 60,138)( 61,142)( 62,143)( 63,144)
( 64,148)( 65,149)( 66,150)( 67,145)( 68,146)( 69,147)( 70,151)( 71,152)
( 72,153)( 73,157)( 74,158)( 75,159)( 76,154)( 77,155)( 78,156)( 79,160)
( 80,161)( 81,162);
s3 := Sym(162)!( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)
( 35, 62)( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)
( 43, 70)( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)
( 51, 78)( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)
(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)
(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)
(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162);
poly := sub<Sym(162)|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, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
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 >;

References : None.
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