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Polytope of Type {2,18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,18}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1857)
Rank : 4
Schlafli Type : {2,18,18}
Number of vertices, edges, etc : 2, 18, 162, 18
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) :
   3-fold quotients : {2,6,18}*432a, {2,18,6}*432a
   9-fold quotients : {2,2,18}*144, {2,18,2}*144, {2,6,6}*144a
   18-fold quotients : {2,2,9}*72, {2,9,2}*72
   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 := (  4,  5)(  7,  8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)( 16, 25)
( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 34, 35)( 37, 38)( 39, 50)
( 40, 49)( 41, 48)( 42, 53)( 43, 52)( 44, 51)( 45, 56)( 46, 55)( 47, 54)
( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)( 68, 75)( 69, 80)( 70, 79)
( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 85, 86)( 88, 89)( 91, 92)( 93,104)
( 94,103)( 95,102)( 96,107)( 97,106)( 98,105)( 99,110)(100,109)(101,108)
(112,113)(115,116)(118,119)(120,131)(121,130)(122,129)(123,134)(124,133)
(125,132)(126,137)(127,136)(128,135)(139,140)(142,143)(145,146)(147,158)
(148,157)(149,156)(150,161)(151,160)(152,159)(153,164)(154,163)(155,162);;
s2 := (  3, 12)(  4, 14)(  5, 13)(  6, 18)(  7, 20)(  8, 19)(  9, 15)( 10, 17)
( 11, 16)( 21, 23)( 24, 29)( 25, 28)( 26, 27)( 30, 72)( 31, 74)( 32, 73)
( 33, 69)( 34, 71)( 35, 70)( 36, 66)( 37, 68)( 38, 67)( 39, 63)( 40, 65)
( 41, 64)( 42, 60)( 43, 62)( 44, 61)( 45, 57)( 46, 59)( 47, 58)( 48, 83)
( 49, 82)( 50, 81)( 51, 80)( 52, 79)( 53, 78)( 54, 77)( 55, 76)( 56, 75)
( 84, 93)( 85, 95)( 86, 94)( 87, 99)( 88,101)( 89,100)( 90, 96)( 91, 98)
( 92, 97)(102,104)(105,110)(106,109)(107,108)(111,153)(112,155)(113,154)
(114,150)(115,152)(116,151)(117,147)(118,149)(119,148)(120,144)(121,146)
(122,145)(123,141)(124,143)(125,142)(126,138)(127,140)(128,139)(129,164)
(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156);;
s3 := (  3,111)(  4,112)(  5,113)(  6,117)(  7,118)(  8,119)(  9,114)( 10,115)
( 11,116)( 12,120)( 13,121)( 14,122)( 15,126)( 16,127)( 17,128)( 18,123)
( 19,124)( 20,125)( 21,129)( 22,130)( 23,131)( 24,135)( 25,136)( 26,137)
( 27,132)( 28,133)( 29,134)( 30, 84)( 31, 85)( 32, 86)( 33, 90)( 34, 91)
( 35, 92)( 36, 87)( 37, 88)( 38, 89)( 39, 93)( 40, 94)( 41, 95)( 42, 99)
( 43,100)( 44,101)( 45, 96)( 46, 97)( 47, 98)( 48,102)( 49,103)( 50,104)
( 51,108)( 52,109)( 53,110)( 54,105)( 55,106)( 56,107)( 57,144)( 58,145)
( 59,146)( 60,141)( 61,142)( 62,143)( 63,138)( 64,139)( 65,140)( 66,153)
( 67,154)( 68,155)( 69,150)( 70,151)( 71,152)( 72,147)( 73,148)( 74,149)
( 75,162)( 76,163)( 77,164)( 78,159)( 79,160)( 80,161)( 81,156)( 82,157)
( 83,158);;
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*s3*s2*s1*s2*s3*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*s1*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, 23)( 13, 22)( 14, 21)( 15, 26)
( 16, 25)( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 34, 35)( 37, 38)
( 39, 50)( 40, 49)( 41, 48)( 42, 53)( 43, 52)( 44, 51)( 45, 56)( 46, 55)
( 47, 54)( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)( 68, 75)( 69, 80)
( 70, 79)( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 85, 86)( 88, 89)( 91, 92)
( 93,104)( 94,103)( 95,102)( 96,107)( 97,106)( 98,105)( 99,110)(100,109)
(101,108)(112,113)(115,116)(118,119)(120,131)(121,130)(122,129)(123,134)
(124,133)(125,132)(126,137)(127,136)(128,135)(139,140)(142,143)(145,146)
(147,158)(148,157)(149,156)(150,161)(151,160)(152,159)(153,164)(154,163)
(155,162);
s2 := Sym(164)!(  3, 12)(  4, 14)(  5, 13)(  6, 18)(  7, 20)(  8, 19)(  9, 15)
( 10, 17)( 11, 16)( 21, 23)( 24, 29)( 25, 28)( 26, 27)( 30, 72)( 31, 74)
( 32, 73)( 33, 69)( 34, 71)( 35, 70)( 36, 66)( 37, 68)( 38, 67)( 39, 63)
( 40, 65)( 41, 64)( 42, 60)( 43, 62)( 44, 61)( 45, 57)( 46, 59)( 47, 58)
( 48, 83)( 49, 82)( 50, 81)( 51, 80)( 52, 79)( 53, 78)( 54, 77)( 55, 76)
( 56, 75)( 84, 93)( 85, 95)( 86, 94)( 87, 99)( 88,101)( 89,100)( 90, 96)
( 91, 98)( 92, 97)(102,104)(105,110)(106,109)(107,108)(111,153)(112,155)
(113,154)(114,150)(115,152)(116,151)(117,147)(118,149)(119,148)(120,144)
(121,146)(122,145)(123,141)(124,143)(125,142)(126,138)(127,140)(128,139)
(129,164)(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)
(137,156);
s3 := Sym(164)!(  3,111)(  4,112)(  5,113)(  6,117)(  7,118)(  8,119)(  9,114)
( 10,115)( 11,116)( 12,120)( 13,121)( 14,122)( 15,126)( 16,127)( 17,128)
( 18,123)( 19,124)( 20,125)( 21,129)( 22,130)( 23,131)( 24,135)( 25,136)
( 26,137)( 27,132)( 28,133)( 29,134)( 30, 84)( 31, 85)( 32, 86)( 33, 90)
( 34, 91)( 35, 92)( 36, 87)( 37, 88)( 38, 89)( 39, 93)( 40, 94)( 41, 95)
( 42, 99)( 43,100)( 44,101)( 45, 96)( 46, 97)( 47, 98)( 48,102)( 49,103)
( 50,104)( 51,108)( 52,109)( 53,110)( 54,105)( 55,106)( 56,107)( 57,144)
( 58,145)( 59,146)( 60,141)( 61,142)( 62,143)( 63,138)( 64,139)( 65,140)
( 66,153)( 67,154)( 68,155)( 69,150)( 70,151)( 71,152)( 72,147)( 73,148)
( 74,149)( 75,162)( 76,163)( 77,164)( 78,159)( 79,160)( 80,161)( 81,156)
( 82,157)( 83,158);
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*s3*s2*s1*s2*s3*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*s1*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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