Polytope of Type {4,82}

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

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