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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}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1860)
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 : {2,6,9}*648c
   3-fold quotients : {2,6,6}*432a
   6-fold quotients : {2,6,3}*216
   9-fold quotients : {2,6,6}*144b
   18-fold quotients : {2,6,3}*72
   27-fold quotients : {2,2,6}*48
   54-fold quotients : {2,2,3}*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, 66)( 31, 68)( 32, 67)( 33, 69)
( 34, 71)( 35, 70)( 36, 72)( 37, 74)( 38, 73)( 39, 57)( 40, 59)( 41, 58)
( 42, 60)( 43, 62)( 44, 61)( 45, 63)( 46, 65)( 47, 64)( 48, 75)( 49, 77)
( 50, 76)( 51, 78)( 52, 80)( 53, 79)( 54, 81)( 55, 83)( 56, 82)( 84, 93)
( 85, 95)( 86, 94)( 87, 96)( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)
(103,104)(106,107)(109,110)(111,147)(112,149)(113,148)(114,150)(115,152)
(116,151)(117,153)(118,155)(119,154)(120,138)(121,140)(122,139)(123,141)
(124,143)(125,142)(126,144)(127,146)(128,145)(129,156)(130,158)(131,157)
(132,159)(133,161)(134,160)(135,162)(136,164)(137,163);;
s3 := (  3,111)(  4,113)(  5,112)(  6,116)(  7,115)(  8,114)(  9,118)( 10,117)
( 11,119)( 12,136)( 13,135)( 14,137)( 15,129)( 16,131)( 17,130)( 18,134)
( 19,133)( 20,132)( 21,123)( 22,125)( 23,124)( 24,128)( 25,127)( 26,126)
( 27,121)( 28,120)( 29,122)( 30, 84)( 31, 86)( 32, 85)( 33, 89)( 34, 88)
( 35, 87)( 36, 91)( 37, 90)( 38, 92)( 39,109)( 40,108)( 41,110)( 42,102)
( 43,104)( 44,103)( 45,107)( 46,106)( 47,105)( 48, 96)( 49, 98)( 50, 97)
( 51,101)( 52,100)( 53, 99)( 54, 94)( 55, 93)( 56, 95)( 57,138)( 58,140)
( 59,139)( 60,143)( 61,142)( 62,141)( 63,145)( 64,144)( 65,146)( 66,163)
( 67,162)( 68,164)( 69,156)( 70,158)( 71,157)( 72,161)( 73,160)( 74,159)
( 75,150)( 76,152)( 77,151)( 78,155)( 79,154)( 80,153)( 81,148)( 82,147)
( 83,149);;
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, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
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(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, 66)( 31, 68)( 32, 67)
( 33, 69)( 34, 71)( 35, 70)( 36, 72)( 37, 74)( 38, 73)( 39, 57)( 40, 59)
( 41, 58)( 42, 60)( 43, 62)( 44, 61)( 45, 63)( 46, 65)( 47, 64)( 48, 75)
( 49, 77)( 50, 76)( 51, 78)( 52, 80)( 53, 79)( 54, 81)( 55, 83)( 56, 82)
( 84, 93)( 85, 95)( 86, 94)( 87, 96)( 88, 98)( 89, 97)( 90, 99)( 91,101)
( 92,100)(103,104)(106,107)(109,110)(111,147)(112,149)(113,148)(114,150)
(115,152)(116,151)(117,153)(118,155)(119,154)(120,138)(121,140)(122,139)
(123,141)(124,143)(125,142)(126,144)(127,146)(128,145)(129,156)(130,158)
(131,157)(132,159)(133,161)(134,160)(135,162)(136,164)(137,163);
s3 := Sym(164)!(  3,111)(  4,113)(  5,112)(  6,116)(  7,115)(  8,114)(  9,118)
( 10,117)( 11,119)( 12,136)( 13,135)( 14,137)( 15,129)( 16,131)( 17,130)
( 18,134)( 19,133)( 20,132)( 21,123)( 22,125)( 23,124)( 24,128)( 25,127)
( 26,126)( 27,121)( 28,120)( 29,122)( 30, 84)( 31, 86)( 32, 85)( 33, 89)
( 34, 88)( 35, 87)( 36, 91)( 37, 90)( 38, 92)( 39,109)( 40,108)( 41,110)
( 42,102)( 43,104)( 44,103)( 45,107)( 46,106)( 47,105)( 48, 96)( 49, 98)
( 50, 97)( 51,101)( 52,100)( 53, 99)( 54, 94)( 55, 93)( 56, 95)( 57,138)
( 58,140)( 59,139)( 60,143)( 61,142)( 62,141)( 63,145)( 64,144)( 65,146)
( 66,163)( 67,162)( 68,164)( 69,156)( 70,158)( 71,157)( 72,161)( 73,160)
( 74,159)( 75,150)( 76,152)( 77,151)( 78,155)( 79,154)( 80,153)( 81,148)
( 82,147)( 83,149);
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, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >; 
 

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