Polytope of Type {82,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {82,4}*656
Also Known As : {82,4|2}. if this polytope has another name.
Group : SmallGroup(656,40)
Rank : 3
Schlafli Type : {82,4}
Number of vertices, edges, etc : 82, 164, 4
Order of s₀s₁s₂ : 164
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {82,4,2} of size 1312
Vertex Figure Of :
   {2,82,4} of size 1312
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {82,2}*328
   4-fold quotients : {41,2}*164
   41-fold quotients : {2,4}*16
   82-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {164,4}*1312, {82,8}*1312
   3-fold covers : {82,12}*1968, {246,4}*1968a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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