Polytope of Type {86,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {86,4}*688
Also Known As : {86,4|2}. if this polytope has another name.
Group : SmallGroup(688,31)
Rank : 3
Schlafli Type : {86,4}
Number of vertices, edges, etc : 86, 172, 4
Order of s₀s₁s₂ : 172
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {86,4,2} of size 1376
Vertex Figure Of :
   {2,86,4} of size 1376
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {86,2}*344
   4-fold quotients : {43,2}*172
   43-fold quotients : {2,4}*16
   86-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {172,4}*1376, {86,8}*1376
Permutation Representation (GAP) :

s0 := (  2, 43)(  3, 42)(  4, 41)(  5, 40)(  6, 39)(  7, 38)(  8, 37)(  9, 36)
( 10, 35)( 11, 34)( 12, 33)( 13, 32)( 14, 31)( 15, 30)( 16, 29)( 17, 28)
( 18, 27)( 19, 26)( 20, 25)( 21, 24)( 22, 23)( 45, 86)( 46, 85)( 47, 84)
( 48, 83)( 49, 82)( 50, 81)( 51, 80)( 52, 79)( 53, 78)( 54, 77)( 55, 76)
( 56, 75)( 57, 74)( 58, 73)( 59, 72)( 60, 71)( 61, 70)( 62, 69)( 63, 68)
( 64, 67)( 65, 66)( 88,129)( 89,128)( 90,127)( 91,126)( 92,125)( 93,124)
( 94,123)( 95,122)( 96,121)( 97,120)( 98,119)( 99,118)(100,117)(101,116)
(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(131,172)
(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)
(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)
(148,155)(149,154)(150,153)(151,152);;
s1 := (  1,  2)(  3, 43)(  4, 42)(  5, 41)(  6, 40)(  7, 39)(  8, 38)(  9, 37)
( 10, 36)( 11, 35)( 12, 34)( 13, 33)( 14, 32)( 15, 31)( 16, 30)( 17, 29)
( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 44, 45)( 46, 86)( 47, 85)
( 48, 84)( 49, 83)( 50, 82)( 51, 81)( 52, 80)( 53, 79)( 54, 78)( 55, 77)
( 56, 76)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 61, 71)( 62, 70)( 63, 69)
( 64, 68)( 65, 67)( 87,131)( 88,130)( 89,172)( 90,171)( 91,170)( 92,169)
( 93,168)( 94,167)( 95,166)( 96,165)( 97,164)( 98,163)( 99,162)(100,161)
(101,160)(102,159)(103,158)(104,157)(105,156)(106,155)(107,154)(108,153)
(109,152)(110,151)(111,150)(112,149)(113,148)(114,147)(115,146)(116,145)
(117,144)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)
(125,136)(126,135)(127,134)(128,133)(129,132);;
s2 := (  1, 87)(  2, 88)(  3, 89)(  4, 90)(  5, 91)(  6, 92)(  7, 93)(  8, 94)
(  9, 95)( 10, 96)( 11, 97)( 12, 98)( 13, 99)( 14,100)( 15,101)( 16,102)
( 17,103)( 18,104)( 19,105)( 20,106)( 21,107)( 22,108)( 23,109)( 24,110)
( 25,111)( 26,112)( 27,113)( 28,114)( 29,115)( 30,116)( 31,117)( 32,118)
( 33,119)( 34,120)( 35,121)( 36,122)( 37,123)( 38,124)( 39,125)( 40,126)
( 41,127)( 42,128)( 43,129)( 44,130)( 45,131)( 46,132)( 47,133)( 48,134)
( 49,135)( 50,136)( 51,137)( 52,138)( 53,139)( 54,140)( 55,141)( 56,142)
( 57,143)( 58,144)( 59,145)( 60,146)( 61,147)( 62,148)( 63,149)( 64,150)
( 65,151)( 66,152)( 67,153)( 68,154)( 69,155)( 70,156)( 71,157)( 72,158)
( 73,159)( 74,160)( 75,161)( 76,162)( 77,163)( 78,164)( 79,165)( 80,166)
( 81,167)( 82,168)( 83,169)( 84,170)( 85,171)( 86,172);;
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*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(172)!(  2, 43)(  3, 42)(  4, 41)(  5, 40)(  6, 39)(  7, 38)(  8, 37)
(  9, 36)( 10, 35)( 11, 34)( 12, 33)( 13, 32)( 14, 31)( 15, 30)( 16, 29)
( 17, 28)( 18, 27)( 19, 26)( 20, 25)( 21, 24)( 22, 23)( 45, 86)( 46, 85)
( 47, 84)( 48, 83)( 49, 82)( 50, 81)( 51, 80)( 52, 79)( 53, 78)( 54, 77)
( 55, 76)( 56, 75)( 57, 74)( 58, 73)( 59, 72)( 60, 71)( 61, 70)( 62, 69)
( 63, 68)( 64, 67)( 65, 66)( 88,129)( 89,128)( 90,127)( 91,126)( 92,125)
( 93,124)( 94,123)( 95,122)( 96,121)( 97,120)( 98,119)( 99,118)(100,117)
(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)
(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)
(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)
(147,156)(148,155)(149,154)(150,153)(151,152);
s1 := Sym(172)!(  1,  2)(  3, 43)(  4, 42)(  5, 41)(  6, 40)(  7, 39)(  8, 38)
(  9, 37)( 10, 36)( 11, 35)( 12, 34)( 13, 33)( 14, 32)( 15, 31)( 16, 30)
( 17, 29)( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 44, 45)( 46, 86)
( 47, 85)( 48, 84)( 49, 83)( 50, 82)( 51, 81)( 52, 80)( 53, 79)( 54, 78)
( 55, 77)( 56, 76)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 61, 71)( 62, 70)
( 63, 69)( 64, 68)( 65, 67)( 87,131)( 88,130)( 89,172)( 90,171)( 91,170)
( 92,169)( 93,168)( 94,167)( 95,166)( 96,165)( 97,164)( 98,163)( 99,162)
(100,161)(101,160)(102,159)(103,158)(104,157)(105,156)(106,155)(107,154)
(108,153)(109,152)(110,151)(111,150)(112,149)(113,148)(114,147)(115,146)
(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)
(124,137)(125,136)(126,135)(127,134)(128,133)(129,132);
s2 := Sym(172)!(  1, 87)(  2, 88)(  3, 89)(  4, 90)(  5, 91)(  6, 92)(  7, 93)
(  8, 94)(  9, 95)( 10, 96)( 11, 97)( 12, 98)( 13, 99)( 14,100)( 15,101)
( 16,102)( 17,103)( 18,104)( 19,105)( 20,106)( 21,107)( 22,108)( 23,109)
( 24,110)( 25,111)( 26,112)( 27,113)( 28,114)( 29,115)( 30,116)( 31,117)
( 32,118)( 33,119)( 34,120)( 35,121)( 36,122)( 37,123)( 38,124)( 39,125)
( 40,126)( 41,127)( 42,128)( 43,129)( 44,130)( 45,131)( 46,132)( 47,133)
( 48,134)( 49,135)( 50,136)( 51,137)( 52,138)( 53,139)( 54,140)( 55,141)
( 56,142)( 57,143)( 58,144)( 59,145)( 60,146)( 61,147)( 62,148)( 63,149)
( 64,150)( 65,151)( 66,152)( 67,153)( 68,154)( 69,155)( 70,156)( 71,157)
( 72,158)( 73,159)( 74,160)( 75,161)( 76,162)( 77,163)( 78,164)( 79,165)
( 80,166)( 81,167)( 82,168)( 83,169)( 84,170)( 85,171)( 86,172);
poly := sub<Sym(172)|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*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope