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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}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 4
Schlafli Type : {2,6,18}
Number of vertices, edges, etc : 2, 18, 162, 54
Order of s0s1s2s3 : 18
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}*648a
   3-fold quotients : {2,6,18}*432b, {2,6,6}*432a
   6-fold quotients : {2,6,9}*216, {2,6,3}*216
   9-fold quotients : {2,2,18}*144, {2,6,6}*144b
   18-fold quotients : {2,2,9}*72, {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 := (  4,  5)(  7,  8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)
( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)( 47, 55)
( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)
( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 85, 86)( 88, 89)( 91, 92)( 93,102)
( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)(101,109)
(112,113)(115,116)(118,119)(120,129)(121,131)(122,130)(123,132)(124,134)
(125,133)(126,135)(127,137)(128,136)(139,140)(142,143)(145,146)(147,156)
(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(154,164)(155,163);;
s2 := (  3, 12)(  4, 13)(  5, 14)(  6, 18)(  7, 19)(  8, 20)(  9, 15)( 10, 16)
( 11, 17)( 24, 27)( 25, 28)( 26, 29)( 30, 72)( 31, 73)( 32, 74)( 33, 69)
( 34, 70)( 35, 71)( 36, 66)( 37, 67)( 38, 68)( 39, 63)( 40, 64)( 41, 65)
( 42, 60)( 43, 61)( 44, 62)( 45, 57)( 46, 58)( 47, 59)( 48, 81)( 49, 82)
( 50, 83)( 51, 78)( 52, 79)( 53, 80)( 54, 75)( 55, 76)( 56, 77)( 84, 93)
( 85, 94)( 86, 95)( 87, 99)( 88,100)( 89,101)( 90, 96)( 91, 97)( 92, 98)
(105,108)(106,109)(107,110)(111,153)(112,154)(113,155)(114,150)(115,151)
(116,152)(117,147)(118,148)(119,149)(120,144)(121,145)(122,146)(123,141)
(124,142)(125,143)(126,138)(127,139)(128,140)(129,162)(130,163)(131,164)
(132,159)(133,160)(134,161)(135,156)(136,157)(137,158);;
s3 := (  3,111)(  4,112)(  5,113)(  6,117)(  7,118)(  8,119)(  9,114)( 10,115)
( 11,116)( 12,131)( 13,129)( 14,130)( 15,137)( 16,135)( 17,136)( 18,134)
( 19,132)( 20,133)( 21,121)( 22,122)( 23,120)( 24,127)( 25,128)( 26,126)
( 27,124)( 28,125)( 29,123)( 30, 84)( 31, 85)( 32, 86)( 33, 90)( 34, 91)
( 35, 92)( 36, 87)( 37, 88)( 38, 89)( 39,104)( 40,102)( 41,103)( 42,110)
( 43,108)( 44,109)( 45,107)( 46,105)( 47,106)( 48, 94)( 49, 95)( 50, 93)
( 51,100)( 52,101)( 53, 99)( 54, 97)( 55, 98)( 56, 96)( 57,144)( 58,145)
( 59,146)( 60,141)( 61,142)( 62,143)( 63,138)( 64,139)( 65,140)( 66,164)
( 67,162)( 68,163)( 69,161)( 70,159)( 71,160)( 72,158)( 73,156)( 74,157)
( 75,154)( 76,155)( 77,153)( 78,151)( 79,152)( 80,150)( 81,148)( 82,149)
( 83,147);;
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, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(164)!(1,2);
s1 := Sym(164)!(  4,  5)(  7,  8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)
( 16, 26)( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)
( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)
( 47, 55)( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)
( 70, 80)( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 85, 86)( 88, 89)( 91, 92)
( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)
(101,109)(112,113)(115,116)(118,119)(120,129)(121,131)(122,130)(123,132)
(124,134)(125,133)(126,135)(127,137)(128,136)(139,140)(142,143)(145,146)
(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(154,164)
(155,163);
s2 := Sym(164)!(  3, 12)(  4, 13)(  5, 14)(  6, 18)(  7, 19)(  8, 20)(  9, 15)
( 10, 16)( 11, 17)( 24, 27)( 25, 28)( 26, 29)( 30, 72)( 31, 73)( 32, 74)
( 33, 69)( 34, 70)( 35, 71)( 36, 66)( 37, 67)( 38, 68)( 39, 63)( 40, 64)
( 41, 65)( 42, 60)( 43, 61)( 44, 62)( 45, 57)( 46, 58)( 47, 59)( 48, 81)
( 49, 82)( 50, 83)( 51, 78)( 52, 79)( 53, 80)( 54, 75)( 55, 76)( 56, 77)
( 84, 93)( 85, 94)( 86, 95)( 87, 99)( 88,100)( 89,101)( 90, 96)( 91, 97)
( 92, 98)(105,108)(106,109)(107,110)(111,153)(112,154)(113,155)(114,150)
(115,151)(116,152)(117,147)(118,148)(119,149)(120,144)(121,145)(122,146)
(123,141)(124,142)(125,143)(126,138)(127,139)(128,140)(129,162)(130,163)
(131,164)(132,159)(133,160)(134,161)(135,156)(136,157)(137,158);
s3 := Sym(164)!(  3,111)(  4,112)(  5,113)(  6,117)(  7,118)(  8,119)(  9,114)
( 10,115)( 11,116)( 12,131)( 13,129)( 14,130)( 15,137)( 16,135)( 17,136)
( 18,134)( 19,132)( 20,133)( 21,121)( 22,122)( 23,120)( 24,127)( 25,128)
( 26,126)( 27,124)( 28,125)( 29,123)( 30, 84)( 31, 85)( 32, 86)( 33, 90)
( 34, 91)( 35, 92)( 36, 87)( 37, 88)( 38, 89)( 39,104)( 40,102)( 41,103)
( 42,110)( 43,108)( 44,109)( 45,107)( 46,105)( 47,106)( 48, 94)( 49, 95)
( 50, 93)( 51,100)( 52,101)( 53, 99)( 54, 97)( 55, 98)( 56, 96)( 57,144)
( 58,145)( 59,146)( 60,141)( 61,142)( 62,143)( 63,138)( 64,139)( 65,140)
( 66,164)( 67,162)( 68,163)( 69,161)( 70,159)( 71,160)( 72,158)( 73,156)
( 74,157)( 75,154)( 76,155)( 77,153)( 78,151)( 79,152)( 80,150)( 81,148)
( 82,149)( 83,147);
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, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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