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