Polytope of Type {4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*1296
Also Known As : {4,4}_(9,9), {4,4}₁₈. if this polytope has another name.
Group : SmallGroup(1296,1813)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 162, 324, 162
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*648
   9-fold quotients : {4,4}*144
   18-fold quotients : {4,4}*72
   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)(  2,121)(  3,126)(  4,122)(  5,124)(  6,120)(  7,125)(  8,118)
(  9,123)( 10,140)( 11,142)( 12,138)( 13,143)( 14,136)( 15,141)( 16,137)
( 17,139)( 18,144)( 19,108)( 20,101)( 21,103)( 22,102)( 23,104)( 24,106)
( 25,105)( 26,107)( 27,100)( 28,146)( 29,148)( 30,153)( 31,149)( 32,151)
( 33,147)( 34,152)( 35,145)( 36,150)( 37, 89)( 38, 82)( 39, 87)( 40, 83)
( 41, 85)( 42, 90)( 43, 86)( 44, 88)( 45, 84)( 46,135)( 47,128)( 48,130)
( 49,129)( 50,131)( 51,133)( 52,132)( 53,134)( 54,127)( 55, 95)( 56, 97)
( 57, 93)( 58, 98)( 59, 91)( 60, 96)( 61, 92)( 62, 94)( 63, 99)( 64,116)
( 65,109)( 66,114)( 67,110)( 68,112)( 69,117)( 70,113)( 71,115)( 72,111)
( 73,162)( 74,155)( 75,157)( 76,156)( 77,158)( 78,160)( 79,159)( 80,161)
( 81,154);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
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)(  2,121)(  3,126)(  4,122)(  5,124)(  6,120)(  7,125)
(  8,118)(  9,123)( 10,140)( 11,142)( 12,138)( 13,143)( 14,136)( 15,141)
( 16,137)( 17,139)( 18,144)( 19,108)( 20,101)( 21,103)( 22,102)( 23,104)
( 24,106)( 25,105)( 26,107)( 27,100)( 28,146)( 29,148)( 30,153)( 31,149)
( 32,151)( 33,147)( 34,152)( 35,145)( 36,150)( 37, 89)( 38, 82)( 39, 87)
( 40, 83)( 41, 85)( 42, 90)( 43, 86)( 44, 88)( 45, 84)( 46,135)( 47,128)
( 48,130)( 49,129)( 50,131)( 51,133)( 52,132)( 53,134)( 54,127)( 55, 95)
( 56, 97)( 57, 93)( 58, 98)( 59, 91)( 60, 96)( 61, 92)( 62, 94)( 63, 99)
( 64,116)( 65,109)( 66,114)( 67,110)( 68,112)( 69,117)( 70,113)( 71,115)
( 72,111)( 73,162)( 74,155)( 75,157)( 76,156)( 77,158)( 78,160)( 79,159)
( 80,161)( 81,154);
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope