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Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1296d
if this polytope has a name.
Group : SmallGroup(1296,2985)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 6, 54, 54, 18
Order of s0s1s2s3 : 6
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 : {2,6,6}*432c, {6,6,6}*432e
   6-fold quotients : {2,3,6}*216
   9-fold quotients : {2,6,6}*144c, {6,6,2}*144a
   18-fold quotients : {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 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)
(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162);;
s1 := (  1,  4)(  2,  5)(  3,  6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)
( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 58)( 29, 59)( 30, 60)( 31, 55)
( 32, 56)( 33, 57)( 34, 61)( 35, 62)( 36, 63)( 37, 76)( 38, 77)( 39, 78)
( 40, 73)( 41, 74)( 42, 75)( 43, 79)( 44, 80)( 45, 81)( 46, 67)( 47, 68)
( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)( 53, 71)( 54, 72)( 82, 85)
( 83, 86)( 84, 87)( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)
( 97,106)( 98,107)( 99,108)(109,139)(110,140)(111,141)(112,136)(113,137)
(114,138)(115,142)(116,143)(117,144)(118,157)(119,158)(120,159)(121,154)
(122,155)(123,156)(124,160)(125,161)(126,162)(127,148)(128,149)(129,150)
(130,145)(131,146)(132,147)(133,151)(134,152)(135,153);;
s2 := (  1,146)(  2,147)(  3,145)(  4,149)(  5,150)(  6,148)(  7,152)(  8,153)
(  9,151)( 10,138)( 11,136)( 12,137)( 13,141)( 14,139)( 15,140)( 16,144)
( 17,142)( 18,143)( 19,154)( 20,155)( 21,156)( 22,157)( 23,158)( 24,159)
( 25,160)( 26,161)( 27,162)( 28,119)( 29,120)( 30,118)( 31,122)( 32,123)
( 33,121)( 34,125)( 35,126)( 36,124)( 37,111)( 38,109)( 39,110)( 40,114)
( 41,112)( 42,113)( 43,117)( 44,115)( 45,116)( 46,127)( 47,128)( 48,129)
( 49,130)( 50,131)( 51,132)( 52,133)( 53,134)( 54,135)( 55, 92)( 56, 93)
( 57, 91)( 58, 95)( 59, 96)( 60, 94)( 61, 98)( 62, 99)( 63, 97)( 64, 84)
( 65, 82)( 66, 83)( 67, 87)( 68, 85)( 69, 86)( 70, 90)( 71, 88)( 72, 89)
( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)( 79,106)( 80,107)
( 81,108);;
s3 := (  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);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(162)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)
(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)
(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162);
s1 := Sym(162)!(  1,  4)(  2,  5)(  3,  6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)
( 14, 20)( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 58)( 29, 59)( 30, 60)
( 31, 55)( 32, 56)( 33, 57)( 34, 61)( 35, 62)( 36, 63)( 37, 76)( 38, 77)
( 39, 78)( 40, 73)( 41, 74)( 42, 75)( 43, 79)( 44, 80)( 45, 81)( 46, 67)
( 47, 68)( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)( 53, 71)( 54, 72)
( 82, 85)( 83, 86)( 84, 87)( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)
( 96,102)( 97,106)( 98,107)( 99,108)(109,139)(110,140)(111,141)(112,136)
(113,137)(114,138)(115,142)(116,143)(117,144)(118,157)(119,158)(120,159)
(121,154)(122,155)(123,156)(124,160)(125,161)(126,162)(127,148)(128,149)
(129,150)(130,145)(131,146)(132,147)(133,151)(134,152)(135,153);
s2 := Sym(162)!(  1,146)(  2,147)(  3,145)(  4,149)(  5,150)(  6,148)(  7,152)
(  8,153)(  9,151)( 10,138)( 11,136)( 12,137)( 13,141)( 14,139)( 15,140)
( 16,144)( 17,142)( 18,143)( 19,154)( 20,155)( 21,156)( 22,157)( 23,158)
( 24,159)( 25,160)( 26,161)( 27,162)( 28,119)( 29,120)( 30,118)( 31,122)
( 32,123)( 33,121)( 34,125)( 35,126)( 36,124)( 37,111)( 38,109)( 39,110)
( 40,114)( 41,112)( 42,113)( 43,117)( 44,115)( 45,116)( 46,127)( 47,128)
( 48,129)( 49,130)( 50,131)( 51,132)( 52,133)( 53,134)( 54,135)( 55, 92)
( 56, 93)( 57, 91)( 58, 95)( 59, 96)( 60, 94)( 61, 98)( 62, 99)( 63, 97)
( 64, 84)( 65, 82)( 66, 83)( 67, 87)( 68, 85)( 69, 86)( 70, 90)( 71, 88)
( 72, 89)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)( 79,106)
( 80,107)( 81,108);
s3 := Sym(162)!(  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);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >; 
 
References : None.
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