Polytope of Type {2,82,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,82,4}*1312
if this polytope has a name.
Group : SmallGroup(1312,182)
Rank : 4
Schlafli Type : {2,82,4}
Number of vertices, edges, etc : 2, 82, 164, 4
Order of s₀s₁s₂s₃ : 164
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,82,2}*656
   4-fold quotients : {2,41,2}*328
   41-fold quotients : {2,2,4}*32
   82-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4, 43)(  5, 42)(  6, 41)(  7, 40)(  8, 39)(  9, 38)( 10, 37)( 11, 36)
( 12, 35)( 13, 34)( 14, 33)( 15, 32)( 16, 31)( 17, 30)( 18, 29)( 19, 28)
( 20, 27)( 21, 26)( 22, 25)( 23, 24)( 45, 84)( 46, 83)( 47, 82)( 48, 81)
( 49, 80)( 50, 79)( 51, 78)( 52, 77)( 53, 76)( 54, 75)( 55, 74)( 56, 73)
( 57, 72)( 58, 71)( 59, 70)( 60, 69)( 61, 68)( 62, 67)( 63, 66)( 64, 65)
( 86,125)( 87,124)( 88,123)( 89,122)( 90,121)( 91,120)( 92,119)( 93,118)
( 94,117)( 95,116)( 96,115)( 97,114)( 98,113)( 99,112)(100,111)(101,110)
(102,109)(103,108)(104,107)(105,106)(127,166)(128,165)(129,164)(130,163)
(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)
(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147);;
s2 := (  3,  4)(  5, 43)(  6, 42)(  7, 41)(  8, 40)(  9, 39)( 10, 38)( 11, 37)
( 12, 36)( 13, 35)( 14, 34)( 15, 33)( 16, 32)( 17, 31)( 18, 30)( 19, 29)
( 20, 28)( 21, 27)( 22, 26)( 23, 25)( 44, 45)( 46, 84)( 47, 83)( 48, 82)
( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 55, 75)( 56, 74)
( 57, 73)( 58, 72)( 59, 71)( 60, 70)( 61, 69)( 62, 68)( 63, 67)( 64, 66)
( 85,127)( 86,126)( 87,166)( 88,165)( 89,164)( 90,163)( 91,162)( 92,161)
( 93,160)( 94,159)( 95,158)( 96,157)( 97,156)( 98,155)( 99,154)(100,153)
(101,152)(102,151)(103,150)(104,149)(105,148)(106,147)(107,146)(108,145)
(109,144)(110,143)(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)
(117,136)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)
(125,128);;
s3 := (  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)
( 83,165)( 84,166);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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(166)!(1,2);
s1 := Sym(166)!(  4, 43)(  5, 42)(  6, 41)(  7, 40)(  8, 39)(  9, 38)( 10, 37)
( 11, 36)( 12, 35)( 13, 34)( 14, 33)( 15, 32)( 16, 31)( 17, 30)( 18, 29)
( 19, 28)( 20, 27)( 21, 26)( 22, 25)( 23, 24)( 45, 84)( 46, 83)( 47, 82)
( 48, 81)( 49, 80)( 50, 79)( 51, 78)( 52, 77)( 53, 76)( 54, 75)( 55, 74)
( 56, 73)( 57, 72)( 58, 71)( 59, 70)( 60, 69)( 61, 68)( 62, 67)( 63, 66)
( 64, 65)( 86,125)( 87,124)( 88,123)( 89,122)( 90,121)( 91,120)( 92,119)
( 93,118)( 94,117)( 95,116)( 96,115)( 97,114)( 98,113)( 99,112)(100,111)
(101,110)(102,109)(103,108)(104,107)(105,106)(127,166)(128,165)(129,164)
(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)
(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)
(146,147);
s2 := Sym(166)!(  3,  4)(  5, 43)(  6, 42)(  7, 41)(  8, 40)(  9, 39)( 10, 38)
( 11, 37)( 12, 36)( 13, 35)( 14, 34)( 15, 33)( 16, 32)( 17, 31)( 18, 30)
( 19, 29)( 20, 28)( 21, 27)( 22, 26)( 23, 25)( 44, 45)( 46, 84)( 47, 83)
( 48, 82)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 55, 75)
( 56, 74)( 57, 73)( 58, 72)( 59, 71)( 60, 70)( 61, 69)( 62, 68)( 63, 67)
( 64, 66)( 85,127)( 86,126)( 87,166)( 88,165)( 89,164)( 90,163)( 91,162)
( 92,161)( 93,160)( 94,159)( 95,158)( 96,157)( 97,156)( 98,155)( 99,154)
(100,153)(101,152)(102,151)(103,150)(104,149)(105,148)(106,147)(107,146)
(108,145)(109,144)(110,143)(111,142)(112,141)(113,140)(114,139)(115,138)
(116,137)(117,136)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)
(124,129)(125,128);
s3 := Sym(166)!(  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)( 83,165)( 84,166);
poly := sub<Sym(166)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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 >;

to this polytope