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Polytope of Type {18,6,6}

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