Polytope of Type {86,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {86,6}*1032
Also Known As : {86,6|2}. if this polytope has another name.
Group : SmallGroup(1032,47)
Rank : 3
Schlafli Type : {86,6}
Number of vertices, edges, etc : 86, 258, 6
Order of s₀s₁s₂ : 258
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   3-fold quotients : {86,2}*344
   6-fold quotients : {43,2}*172
   43-fold quotients : {2,6}*24
   86-fold quotients : {2,3}*12
   129-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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)(174,215)(175,214)(176,213)(177,212)
(178,211)(179,210)(180,209)(181,208)(182,207)(183,206)(184,205)(185,204)
(186,203)(187,202)(188,201)(189,200)(190,199)(191,198)(192,197)(193,196)
(194,195)(217,258)(218,257)(219,256)(220,255)(221,254)(222,253)(223,252)
(224,251)(225,250)(226,249)(227,248)(228,247)(229,246)(230,245)(231,244)
(232,243)(233,242)(234,241)(235,240)(236,239)(237,238);;
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, 88)( 45, 87)( 46,129)
( 47,128)( 48,127)( 49,126)( 50,125)( 51,124)( 52,123)( 53,122)( 54,121)
( 55,120)( 56,119)( 57,118)( 58,117)( 59,116)( 60,115)( 61,114)( 62,113)
( 63,112)( 64,111)( 65,110)( 66,109)( 67,108)( 68,107)( 69,106)( 70,105)
( 71,104)( 72,103)( 73,102)( 74,101)( 75,100)( 76, 99)( 77, 98)( 78, 97)
( 79, 96)( 80, 95)( 81, 94)( 82, 93)( 83, 92)( 84, 91)( 85, 90)( 86, 89)
(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)(173,217)(174,216)(175,258)
(176,257)(177,256)(178,255)(179,254)(180,253)(181,252)(182,251)(183,250)
(184,249)(185,248)(186,247)(187,246)(188,245)(189,244)(190,243)(191,242)
(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)
(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)
(208,225)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218);;
s2 := (  1,173)(  2,174)(  3,175)(  4,176)(  5,177)(  6,178)(  7,179)(  8,180)
(  9,181)( 10,182)( 11,183)( 12,184)( 13,185)( 14,186)( 15,187)( 16,188)
( 17,189)( 18,190)( 19,191)( 20,192)( 21,193)( 22,194)( 23,195)( 24,196)
( 25,197)( 26,198)( 27,199)( 28,200)( 29,201)( 30,202)( 31,203)( 32,204)
( 33,205)( 34,206)( 35,207)( 36,208)( 37,209)( 38,210)( 39,211)( 40,212)
( 41,213)( 42,214)( 43,215)( 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)( 87,216)( 88,217)
( 89,218)( 90,219)( 91,220)( 92,221)( 93,222)( 94,223)( 95,224)( 96,225)
( 97,226)( 98,227)( 99,228)(100,229)(101,230)(102,231)(103,232)(104,233)
(105,234)(106,235)(107,236)(108,237)(109,238)(110,239)(111,240)(112,241)
(113,242)(114,243)(115,244)(116,245)(117,246)(118,247)(119,248)(120,249)
(121,250)(122,251)(123,252)(124,253)(125,254)(126,255)(127,256)(128,257)
(129,258);;
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*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(258)!(  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)(174,215)(175,214)(176,213)
(177,212)(178,211)(179,210)(180,209)(181,208)(182,207)(183,206)(184,205)
(185,204)(186,203)(187,202)(188,201)(189,200)(190,199)(191,198)(192,197)
(193,196)(194,195)(217,258)(218,257)(219,256)(220,255)(221,254)(222,253)
(223,252)(224,251)(225,250)(226,249)(227,248)(228,247)(229,246)(230,245)
(231,244)(232,243)(233,242)(234,241)(235,240)(236,239)(237,238);
s1 := Sym(258)!(  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, 88)( 45, 87)
( 46,129)( 47,128)( 48,127)( 49,126)( 50,125)( 51,124)( 52,123)( 53,122)
( 54,121)( 55,120)( 56,119)( 57,118)( 58,117)( 59,116)( 60,115)( 61,114)
( 62,113)( 63,112)( 64,111)( 65,110)( 66,109)( 67,108)( 68,107)( 69,106)
( 70,105)( 71,104)( 72,103)( 73,102)( 74,101)( 75,100)( 76, 99)( 77, 98)
( 78, 97)( 79, 96)( 80, 95)( 81, 94)( 82, 93)( 83, 92)( 84, 91)( 85, 90)
( 86, 89)(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)(173,217)(174,216)
(175,258)(176,257)(177,256)(178,255)(179,254)(180,253)(181,252)(182,251)
(183,250)(184,249)(185,248)(186,247)(187,246)(188,245)(189,244)(190,243)
(191,242)(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)
(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)
(207,226)(208,225)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)
(215,218);
s2 := Sym(258)!(  1,173)(  2,174)(  3,175)(  4,176)(  5,177)(  6,178)(  7,179)
(  8,180)(  9,181)( 10,182)( 11,183)( 12,184)( 13,185)( 14,186)( 15,187)
( 16,188)( 17,189)( 18,190)( 19,191)( 20,192)( 21,193)( 22,194)( 23,195)
( 24,196)( 25,197)( 26,198)( 27,199)( 28,200)( 29,201)( 30,202)( 31,203)
( 32,204)( 33,205)( 34,206)( 35,207)( 36,208)( 37,209)( 38,210)( 39,211)
( 40,212)( 41,213)( 42,214)( 43,215)( 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)( 87,216)
( 88,217)( 89,218)( 90,219)( 91,220)( 92,221)( 93,222)( 94,223)( 95,224)
( 96,225)( 97,226)( 98,227)( 99,228)(100,229)(101,230)(102,231)(103,232)
(104,233)(105,234)(106,235)(107,236)(108,237)(109,238)(110,239)(111,240)
(112,241)(113,242)(114,243)(115,244)(116,245)(117,246)(118,247)(119,248)
(120,249)(121,250)(122,251)(123,252)(124,253)(125,254)(126,255)(127,256)
(128,257)(129,258);
poly := sub<Sym(258)|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*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