Polytope of Type {4,86}

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

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

Permutation Representation (Magma) :

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

References : None.
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