Polytope of Type {18,4}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope