Polytope of Type {2,86,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,86,4}*1376
if this polytope has a name.
Group : SmallGroup(1376,182)
Rank : 4
Schlafli Type : {2,86,4}
Number of vertices, edges, etc : 2, 86, 172, 4
Order of s₀s₁s₂s₃ : 172
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 : {2,86,2}*688
   4-fold quotients : {2,43,2}*344
   43-fold quotients : {2,2,4}*32
   86-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4, 45)(  5, 44)(  6, 43)(  7, 42)(  8, 41)(  9, 40)( 10, 39)( 11, 38)
( 12, 37)( 13, 36)( 14, 35)( 15, 34)( 16, 33)( 17, 32)( 18, 31)( 19, 30)
( 20, 29)( 21, 28)( 22, 27)( 23, 26)( 24, 25)( 47, 88)( 48, 87)( 49, 86)
( 50, 85)( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)( 56, 79)( 57, 78)
( 58, 77)( 59, 76)( 60, 75)( 61, 74)( 62, 73)( 63, 72)( 64, 71)( 65, 70)
( 66, 69)( 67, 68)( 90,131)( 91,130)( 92,129)( 93,128)( 94,127)( 95,126)
( 96,125)( 97,124)( 98,123)( 99,122)(100,121)(101,120)(102,119)(103,118)
(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(133,174)
(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)
(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)
(150,157)(151,156)(152,155)(153,154);;
s2 := (  3,  4)(  5, 45)(  6, 44)(  7, 43)(  8, 42)(  9, 41)( 10, 40)( 11, 39)
( 12, 38)( 13, 37)( 14, 36)( 15, 35)( 16, 34)( 17, 33)( 18, 32)( 19, 31)
( 20, 30)( 21, 29)( 22, 28)( 23, 27)( 24, 26)( 46, 47)( 48, 88)( 49, 87)
( 50, 86)( 51, 85)( 52, 84)( 53, 83)( 54, 82)( 55, 81)( 56, 80)( 57, 79)
( 58, 78)( 59, 77)( 60, 76)( 61, 75)( 62, 74)( 63, 73)( 64, 72)( 65, 71)
( 66, 70)( 67, 69)( 89,133)( 90,132)( 91,174)( 92,173)( 93,172)( 94,171)
( 95,170)( 96,169)( 97,168)( 98,167)( 99,166)(100,165)(101,164)(102,163)
(103,162)(104,161)(105,160)(106,159)(107,158)(108,157)(109,156)(110,155)
(111,154)(112,153)(113,152)(114,151)(115,150)(116,149)(117,148)(118,147)
(119,146)(120,145)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)
(127,138)(128,137)(129,136)(130,135)(131,134);;
s3 := (  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)( 87,173)( 88,174);;
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*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(174)!(1,2);
s1 := Sym(174)!(  4, 45)(  5, 44)(  6, 43)(  7, 42)(  8, 41)(  9, 40)( 10, 39)
( 11, 38)( 12, 37)( 13, 36)( 14, 35)( 15, 34)( 16, 33)( 17, 32)( 18, 31)
( 19, 30)( 20, 29)( 21, 28)( 22, 27)( 23, 26)( 24, 25)( 47, 88)( 48, 87)
( 49, 86)( 50, 85)( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)( 56, 79)
( 57, 78)( 58, 77)( 59, 76)( 60, 75)( 61, 74)( 62, 73)( 63, 72)( 64, 71)
( 65, 70)( 66, 69)( 67, 68)( 90,131)( 91,130)( 92,129)( 93,128)( 94,127)
( 95,126)( 96,125)( 97,124)( 98,123)( 99,122)(100,121)(101,120)(102,119)
(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)
(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)
(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)
(149,158)(150,157)(151,156)(152,155)(153,154);
s2 := Sym(174)!(  3,  4)(  5, 45)(  6, 44)(  7, 43)(  8, 42)(  9, 41)( 10, 40)
( 11, 39)( 12, 38)( 13, 37)( 14, 36)( 15, 35)( 16, 34)( 17, 33)( 18, 32)
( 19, 31)( 20, 30)( 21, 29)( 22, 28)( 23, 27)( 24, 26)( 46, 47)( 48, 88)
( 49, 87)( 50, 86)( 51, 85)( 52, 84)( 53, 83)( 54, 82)( 55, 81)( 56, 80)
( 57, 79)( 58, 78)( 59, 77)( 60, 76)( 61, 75)( 62, 74)( 63, 73)( 64, 72)
( 65, 71)( 66, 70)( 67, 69)( 89,133)( 90,132)( 91,174)( 92,173)( 93,172)
( 94,171)( 95,170)( 96,169)( 97,168)( 98,167)( 99,166)(100,165)(101,164)
(102,163)(103,162)(104,161)(105,160)(106,159)(107,158)(108,157)(109,156)
(110,155)(111,154)(112,153)(113,152)(114,151)(115,150)(116,149)(117,148)
(118,147)(119,146)(120,145)(121,144)(122,143)(123,142)(124,141)(125,140)
(126,139)(127,138)(128,137)(129,136)(130,135)(131,134);
s3 := Sym(174)!(  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)( 87,173)( 88,174);
poly := sub<Sym(174)|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*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope