Polytope of Type {2,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,6}*1296g
if this polytope has a name.
Group : SmallGroup(1296,1862)
Rank : 4
Schlafli Type : {2,18,6}
Number of vertices, edges, etc : 2, 54, 162, 18
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 : {2,18,6}*648c
   3-fold quotients : {2,6,6}*432b
   6-fold quotients : {2,6,6}*216
   9-fold quotients : {2,6,6}*144a
   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, 59)( 31, 58)( 32, 57)( 33, 65)
( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 68)( 40, 67)( 41, 66)
( 42, 74)( 43, 73)( 44, 72)( 45, 71)( 46, 70)( 47, 69)( 48, 77)( 49, 76)
( 50, 75)( 51, 83)( 52, 82)( 53, 81)( 54, 80)( 55, 79)( 56, 78)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,140)(112,139)(113,138)(114,146)(115,145)
(116,144)(117,143)(118,142)(119,141)(120,149)(121,148)(122,147)(123,155)
(124,154)(125,153)(126,152)(127,151)(128,150)(129,158)(130,157)(131,156)
(132,164)(133,163)(134,162)(135,161)(136,160)(137,159);;
s2 := (  3, 30)(  4, 32)(  5, 31)(  6, 35)(  7, 34)(  8, 33)(  9, 37)( 10, 36)
( 11, 38)( 12, 55)( 13, 54)( 14, 56)( 15, 48)( 16, 50)( 17, 49)( 18, 53)
( 19, 52)( 20, 51)( 21, 42)( 22, 44)( 23, 43)( 24, 47)( 25, 46)( 26, 45)
( 27, 40)( 28, 39)( 29, 41)( 57, 59)( 60, 61)( 64, 65)( 66, 81)( 67, 83)
( 68, 82)( 69, 77)( 70, 76)( 71, 75)( 72, 79)( 73, 78)( 74, 80)( 84,111)
( 85,113)( 86,112)( 87,116)( 88,115)( 89,114)( 90,118)( 91,117)( 92,119)
( 93,136)( 94,135)( 95,137)( 96,129)( 97,131)( 98,130)( 99,134)(100,133)
(101,132)(102,123)(103,125)(104,124)(105,128)(106,127)(107,126)(108,121)
(109,120)(110,122)(138,140)(141,142)(145,146)(147,162)(148,164)(149,163)
(150,158)(151,157)(152,156)(153,160)(154,159)(155,161);;
s3 := (  3, 93)(  4, 94)(  5, 95)(  6, 99)(  7,100)(  8,101)(  9, 96)( 10, 97)
( 11, 98)( 12, 84)( 13, 85)( 14, 86)( 15, 90)( 16, 91)( 17, 92)( 18, 87)
( 19, 88)( 20, 89)( 21,102)( 22,103)( 23,104)( 24,108)( 25,109)( 26,110)
( 27,105)( 28,106)( 29,107)( 30,120)( 31,121)( 32,122)( 33,126)( 34,127)
( 35,128)( 36,123)( 37,124)( 38,125)( 39,111)( 40,112)( 41,113)( 42,117)
( 43,118)( 44,119)( 45,114)( 46,115)( 47,116)( 48,129)( 49,130)( 50,131)
( 51,135)( 52,136)( 53,137)( 54,132)( 55,133)( 56,134)( 57,147)( 58,148)
( 59,149)( 60,153)( 61,154)( 62,155)( 63,150)( 64,151)( 65,152)( 66,138)
( 67,139)( 68,140)( 69,144)( 70,145)( 71,146)( 72,141)( 73,142)( 74,143)
( 75,156)( 76,157)( 77,158)( 78,162)( 79,163)( 80,164)( 81,159)( 82,160)
( 83,161);;
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, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*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, 59)( 31, 58)( 32, 57)
( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 68)( 40, 67)
( 41, 66)( 42, 74)( 43, 73)( 44, 72)( 45, 71)( 46, 70)( 47, 69)( 48, 77)
( 49, 76)( 50, 75)( 51, 83)( 52, 82)( 53, 81)( 54, 80)( 55, 79)( 56, 78)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,140)(112,139)(113,138)(114,146)
(115,145)(116,144)(117,143)(118,142)(119,141)(120,149)(121,148)(122,147)
(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,158)(130,157)
(131,156)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159);
s2 := Sym(164)!(  3, 30)(  4, 32)(  5, 31)(  6, 35)(  7, 34)(  8, 33)(  9, 37)
( 10, 36)( 11, 38)( 12, 55)( 13, 54)( 14, 56)( 15, 48)( 16, 50)( 17, 49)
( 18, 53)( 19, 52)( 20, 51)( 21, 42)( 22, 44)( 23, 43)( 24, 47)( 25, 46)
( 26, 45)( 27, 40)( 28, 39)( 29, 41)( 57, 59)( 60, 61)( 64, 65)( 66, 81)
( 67, 83)( 68, 82)( 69, 77)( 70, 76)( 71, 75)( 72, 79)( 73, 78)( 74, 80)
( 84,111)( 85,113)( 86,112)( 87,116)( 88,115)( 89,114)( 90,118)( 91,117)
( 92,119)( 93,136)( 94,135)( 95,137)( 96,129)( 97,131)( 98,130)( 99,134)
(100,133)(101,132)(102,123)(103,125)(104,124)(105,128)(106,127)(107,126)
(108,121)(109,120)(110,122)(138,140)(141,142)(145,146)(147,162)(148,164)
(149,163)(150,158)(151,157)(152,156)(153,160)(154,159)(155,161);
s3 := Sym(164)!(  3, 93)(  4, 94)(  5, 95)(  6, 99)(  7,100)(  8,101)(  9, 96)
( 10, 97)( 11, 98)( 12, 84)( 13, 85)( 14, 86)( 15, 90)( 16, 91)( 17, 92)
( 18, 87)( 19, 88)( 20, 89)( 21,102)( 22,103)( 23,104)( 24,108)( 25,109)
( 26,110)( 27,105)( 28,106)( 29,107)( 30,120)( 31,121)( 32,122)( 33,126)
( 34,127)( 35,128)( 36,123)( 37,124)( 38,125)( 39,111)( 40,112)( 41,113)
( 42,117)( 43,118)( 44,119)( 45,114)( 46,115)( 47,116)( 48,129)( 49,130)
( 50,131)( 51,135)( 52,136)( 53,137)( 54,132)( 55,133)( 56,134)( 57,147)
( 58,148)( 59,149)( 60,153)( 61,154)( 62,155)( 63,150)( 64,151)( 65,152)
( 66,138)( 67,139)( 68,140)( 69,144)( 70,145)( 71,146)( 72,141)( 73,142)
( 74,143)( 75,156)( 76,157)( 77,158)( 78,162)( 79,163)( 80,164)( 81,159)
( 82,160)( 83,161);
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, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope