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Polytope of Type {2,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,18}*1296g
if this polytope has a name.
Group : SmallGroup(1296,1862)
Rank : 4
Schlafli Type : {2,6,18}
Number of vertices, edges, etc : 2, 18, 162, 54
Order of s0s1s2s3 : 6
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 : {2,6,18}*648c
   3-fold quotients : {2,6,6}*432b
   6-fold quotients : {2,6,6}*216
   9-fold quotients : {2,6,6}*144a
   27-fold quotients : {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {2,2,3}*24, {2,3,2}*24
   81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  6, 10)(  7, 11)(  8,  9)( 12, 21)( 13, 22)( 14, 23)( 15, 28)( 16, 29)
( 17, 27)( 18, 26)( 19, 24)( 20, 25)( 33, 37)( 34, 38)( 35, 36)( 39, 48)
( 40, 49)( 41, 50)( 42, 55)( 43, 56)( 44, 54)( 45, 53)( 46, 51)( 47, 52)
( 60, 64)( 61, 65)( 62, 63)( 66, 75)( 67, 76)( 68, 77)( 69, 82)( 70, 83)
( 71, 81)( 72, 80)( 73, 78)( 74, 79)( 87, 91)( 88, 92)( 89, 90)( 93,102)
( 94,103)( 95,104)( 96,109)( 97,110)( 98,108)( 99,107)(100,105)(101,106)
(114,118)(115,119)(116,117)(120,129)(121,130)(122,131)(123,136)(124,137)
(125,135)(126,134)(127,132)(128,133)(141,145)(142,146)(143,144)(147,156)
(148,157)(149,158)(150,163)(151,164)(152,162)(153,161)(154,159)(155,160);;
s2 := (  3, 12)(  4, 14)(  5, 13)(  6, 15)(  7, 17)(  8, 16)(  9, 18)( 10, 20)
( 11, 19)( 22, 23)( 25, 26)( 28, 29)( 30, 68)( 31, 67)( 32, 66)( 33, 71)
( 34, 70)( 35, 69)( 36, 74)( 37, 73)( 38, 72)( 39, 59)( 40, 58)( 41, 57)
( 42, 62)( 43, 61)( 44, 60)( 45, 65)( 46, 64)( 47, 63)( 48, 77)( 49, 76)
( 50, 75)( 51, 80)( 52, 79)( 53, 78)( 54, 83)( 55, 82)( 56, 81)( 84, 93)
( 85, 95)( 86, 94)( 87, 96)( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)
(103,104)(106,107)(109,110)(111,149)(112,148)(113,147)(114,152)(115,151)
(116,150)(117,155)(118,154)(119,153)(120,140)(121,139)(122,138)(123,143)
(124,142)(125,141)(126,146)(127,145)(128,144)(129,158)(130,157)(131,156)
(132,161)(133,160)(134,159)(135,164)(136,163)(137,162);;
s3 := (  3,111)(  4,113)(  5,112)(  6,117)(  7,119)(  8,118)(  9,114)( 10,116)
( 11,115)( 12,123)( 13,125)( 14,124)( 15,120)( 16,122)( 17,121)( 18,126)
( 19,128)( 20,127)( 21,136)( 22,135)( 23,137)( 24,133)( 25,132)( 26,134)
( 27,130)( 28,129)( 29,131)( 30, 84)( 31, 86)( 32, 85)( 33, 90)( 34, 92)
( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39, 96)( 40, 98)( 41, 97)( 42, 93)
( 43, 95)( 44, 94)( 45, 99)( 46,101)( 47,100)( 48,109)( 49,108)( 50,110)
( 51,106)( 52,105)( 53,107)( 54,103)( 55,102)( 56,104)( 57,140)( 58,139)
( 59,138)( 60,146)( 61,145)( 62,144)( 63,143)( 64,142)( 65,141)( 66,152)
( 67,151)( 68,150)( 69,149)( 70,148)( 71,147)( 72,155)( 73,154)( 74,153)
( 75,162)( 76,164)( 77,163)( 78,159)( 79,161)( 80,160)( 81,156)( 82,158)
( 83,157);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(164)!(1,2);
s1 := Sym(164)!(  6, 10)(  7, 11)(  8,  9)( 12, 21)( 13, 22)( 14, 23)( 15, 28)
( 16, 29)( 17, 27)( 18, 26)( 19, 24)( 20, 25)( 33, 37)( 34, 38)( 35, 36)
( 39, 48)( 40, 49)( 41, 50)( 42, 55)( 43, 56)( 44, 54)( 45, 53)( 46, 51)
( 47, 52)( 60, 64)( 61, 65)( 62, 63)( 66, 75)( 67, 76)( 68, 77)( 69, 82)
( 70, 83)( 71, 81)( 72, 80)( 73, 78)( 74, 79)( 87, 91)( 88, 92)( 89, 90)
( 93,102)( 94,103)( 95,104)( 96,109)( 97,110)( 98,108)( 99,107)(100,105)
(101,106)(114,118)(115,119)(116,117)(120,129)(121,130)(122,131)(123,136)
(124,137)(125,135)(126,134)(127,132)(128,133)(141,145)(142,146)(143,144)
(147,156)(148,157)(149,158)(150,163)(151,164)(152,162)(153,161)(154,159)
(155,160);
s2 := Sym(164)!(  3, 12)(  4, 14)(  5, 13)(  6, 15)(  7, 17)(  8, 16)(  9, 18)
( 10, 20)( 11, 19)( 22, 23)( 25, 26)( 28, 29)( 30, 68)( 31, 67)( 32, 66)
( 33, 71)( 34, 70)( 35, 69)( 36, 74)( 37, 73)( 38, 72)( 39, 59)( 40, 58)
( 41, 57)( 42, 62)( 43, 61)( 44, 60)( 45, 65)( 46, 64)( 47, 63)( 48, 77)
( 49, 76)( 50, 75)( 51, 80)( 52, 79)( 53, 78)( 54, 83)( 55, 82)( 56, 81)
( 84, 93)( 85, 95)( 86, 94)( 87, 96)( 88, 98)( 89, 97)( 90, 99)( 91,101)
( 92,100)(103,104)(106,107)(109,110)(111,149)(112,148)(113,147)(114,152)
(115,151)(116,150)(117,155)(118,154)(119,153)(120,140)(121,139)(122,138)
(123,143)(124,142)(125,141)(126,146)(127,145)(128,144)(129,158)(130,157)
(131,156)(132,161)(133,160)(134,159)(135,164)(136,163)(137,162);
s3 := Sym(164)!(  3,111)(  4,113)(  5,112)(  6,117)(  7,119)(  8,118)(  9,114)
( 10,116)( 11,115)( 12,123)( 13,125)( 14,124)( 15,120)( 16,122)( 17,121)
( 18,126)( 19,128)( 20,127)( 21,136)( 22,135)( 23,137)( 24,133)( 25,132)
( 26,134)( 27,130)( 28,129)( 29,131)( 30, 84)( 31, 86)( 32, 85)( 33, 90)
( 34, 92)( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39, 96)( 40, 98)( 41, 97)
( 42, 93)( 43, 95)( 44, 94)( 45, 99)( 46,101)( 47,100)( 48,109)( 49,108)
( 50,110)( 51,106)( 52,105)( 53,107)( 54,103)( 55,102)( 56,104)( 57,140)
( 58,139)( 59,138)( 60,146)( 61,145)( 62,144)( 63,143)( 64,142)( 65,141)
( 66,152)( 67,151)( 68,150)( 69,149)( 70,148)( 71,147)( 72,155)( 73,154)
( 74,153)( 75,162)( 76,164)( 77,163)( 78,159)( 79,161)( 80,160)( 81,156)
( 82,158)( 83,157);
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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >; 
 

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