Polytope of Type {2,18,6}

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

Permutation Representation (Magma) :

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

to this polytope