Polytope of Type {4,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18}*1296a
Also Known As : {4,18}₄. if this polytope has another name.
Group : SmallGroup(1296,1813)
Rank : 3
Schlafli Type : {4,18}
Number of vertices, edges, etc : 36, 324, 162
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,18}*648
   9-fold quotients : {4,6}*144
   18-fold quotients : {4,6}*72
   81-fold quotients : {4,2}*16
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
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 >;

References : None.
to this polytope