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Polytope of Type {6,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18,6}*1296a
Also Known As : {{6,18|2},{18,6|2}}. if this polytope has another name.
Group : SmallGroup(1296,2984)
Rank : 4
Schlafli Type : {6,18,6}
Number of vertices, edges, etc : 6, 54, 54, 6
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
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 : {2,18,6}*432a, {6,18,2}*432a, {6,6,6}*432b
   9-fold quotients : {2,18,2}*144, {2,6,6}*144a, {6,2,6}*144, {6,6,2}*144a
   18-fold quotients : {2,9,2}*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 := ( 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);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)(  8, 36)
(  9, 35)( 10, 47)( 11, 46)( 12, 48)( 13, 50)( 14, 49)( 15, 51)( 16, 53)
( 17, 52)( 18, 54)( 19, 38)( 20, 37)( 21, 39)( 22, 41)( 23, 40)( 24, 42)
( 25, 44)( 26, 43)( 27, 45)( 56, 57)( 59, 60)( 62, 63)( 64, 74)( 65, 73)
( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 82,109)
( 83,111)( 84,110)( 85,112)( 86,114)( 87,113)( 88,115)( 89,117)( 90,116)
( 91,128)( 92,127)( 93,129)( 94,131)( 95,130)( 96,132)( 97,134)( 98,133)
( 99,135)(100,119)(101,118)(102,120)(103,122)(104,121)(105,123)(106,125)
(107,124)(108,126)(137,138)(140,141)(143,144)(145,155)(146,154)(147,156)
(148,158)(149,157)(150,159)(151,161)(152,160)(153,162);;
s2 := (  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, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 47)( 49, 53)
( 50, 52)( 51, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 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)(109,118)(110,120)(111,119)(112,124)
(113,126)(114,125)(115,121)(116,123)(117,122)(127,128)(130,134)(131,133)
(132,135)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)(142,148)
(143,150)(144,149)(154,155)(157,161)(158,160)(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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(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);
s1 := Sym(162)!(  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)
(  8, 36)(  9, 35)( 10, 47)( 11, 46)( 12, 48)( 13, 50)( 14, 49)( 15, 51)
( 16, 53)( 17, 52)( 18, 54)( 19, 38)( 20, 37)( 21, 39)( 22, 41)( 23, 40)
( 24, 42)( 25, 44)( 26, 43)( 27, 45)( 56, 57)( 59, 60)( 62, 63)( 64, 74)
( 65, 73)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)
( 82,109)( 83,111)( 84,110)( 85,112)( 86,114)( 87,113)( 88,115)( 89,117)
( 90,116)( 91,128)( 92,127)( 93,129)( 94,131)( 95,130)( 96,132)( 97,134)
( 98,133)( 99,135)(100,119)(101,118)(102,120)(103,122)(104,121)(105,123)
(106,125)(107,124)(108,126)(137,138)(140,141)(143,144)(145,155)(146,154)
(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162);
s2 := 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, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 47)
( 49, 53)( 50, 52)( 51, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 74)( 76, 80)( 77, 79)( 78, 81)
( 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)(109,118)(110,120)(111,119)
(112,124)(113,126)(114,125)(115,121)(116,123)(117,122)(127,128)(130,134)
(131,133)(132,135)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)
(142,148)(143,150)(144,149)(154,155)(157,161)(158,160)(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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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 >; 
 
References : None.
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