Polytope of Type {6,86}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,86}*1032
Also Known As : {6,86|2}. if this polytope has another name.
Group : SmallGroup(1032,47)
Rank : 3
Schlafli Type : {6,86}
Number of vertices, edges, etc : 6, 258, 86
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 : {2,86}*344
   6-fold quotients : {2,43}*172
   43-fold quotients : {6,2}*24
   86-fold quotients : {3,2}*12
   129-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 44, 87)( 45, 88)( 46, 89)( 47, 90)( 48, 91)( 49, 92)( 50, 93)( 51, 94)
( 52, 95)( 53, 96)( 54, 97)( 55, 98)( 56, 99)( 57,100)( 58,101)( 59,102)
( 60,103)( 61,104)( 62,105)( 63,106)( 64,107)( 65,108)( 66,109)( 67,110)
( 68,111)( 69,112)( 70,113)( 71,114)( 72,115)( 73,116)( 74,117)( 75,118)
( 76,119)( 77,120)( 78,121)( 79,122)( 80,123)( 81,124)( 82,125)( 83,126)
( 84,127)( 85,128)( 86,129)(173,216)(174,217)(175,218)(176,219)(177,220)
(178,221)(179,222)(180,223)(181,224)(182,225)(183,226)(184,227)(185,228)
(186,229)(187,230)(188,231)(189,232)(190,233)(191,234)(192,235)(193,236)
(194,237)(195,238)(196,239)(197,240)(198,241)(199,242)(200,243)(201,244)
(202,245)(203,246)(204,247)(205,248)(206,249)(207,250)(208,251)(209,252)
(210,253)(211,254)(212,255)(213,256)(214,257)(215,258);;
s1 := (  1, 44)(  2, 86)(  3, 85)(  4, 84)(  5, 83)(  6, 82)(  7, 81)(  8, 80)
(  9, 79)( 10, 78)( 11, 77)( 12, 76)( 13, 75)( 14, 74)( 15, 73)( 16, 72)
( 17, 71)( 18, 70)( 19, 69)( 20, 68)( 21, 67)( 22, 66)( 23, 65)( 24, 64)
( 25, 63)( 26, 62)( 27, 61)( 28, 60)( 29, 59)( 30, 58)( 31, 57)( 32, 56)
( 33, 55)( 34, 54)( 35, 53)( 36, 52)( 37, 51)( 38, 50)( 39, 49)( 40, 48)
( 41, 47)( 42, 46)( 43, 45)( 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)
(130,173)(131,215)(132,214)(133,213)(134,212)(135,211)(136,210)(137,209)
(138,208)(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)(145,201)
(146,200)(147,199)(148,198)(149,197)(150,196)(151,195)(152,194)(153,193)
(154,192)(155,191)(156,190)(157,189)(158,188)(159,187)(160,186)(161,185)
(162,184)(163,183)(164,182)(165,181)(166,180)(167,179)(168,178)(169,177)
(170,176)(171,175)(172,174)(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);;
s2 := (  1,131)(  2,130)(  3,172)(  4,171)(  5,170)(  6,169)(  7,168)(  8,167)
(  9,166)( 10,165)( 11,164)( 12,163)( 13,162)( 14,161)( 15,160)( 16,159)
( 17,158)( 18,157)( 19,156)( 20,155)( 21,154)( 22,153)( 23,152)( 24,151)
( 25,150)( 26,149)( 27,148)( 28,147)( 29,146)( 30,145)( 31,144)( 32,143)
( 33,142)( 34,141)( 35,140)( 36,139)( 37,138)( 38,137)( 39,136)( 40,135)
( 41,134)( 42,133)( 43,132)( 44,174)( 45,173)( 46,215)( 47,214)( 48,213)
( 49,212)( 50,211)( 51,210)( 52,209)( 53,208)( 54,207)( 55,206)( 56,205)
( 57,204)( 58,203)( 59,202)( 60,201)( 61,200)( 62,199)( 63,198)( 64,197)
( 65,196)( 66,195)( 67,194)( 68,193)( 69,192)( 70,191)( 71,190)( 72,189)
( 73,188)( 74,187)( 75,186)( 76,185)( 77,184)( 78,183)( 79,182)( 80,181)
( 81,180)( 82,179)( 83,178)( 84,177)( 85,176)( 86,175)( 87,217)( 88,216)
( 89,258)( 90,257)( 91,256)( 92,255)( 93,254)( 94,253)( 95,252)( 96,251)
( 97,250)( 98,249)( 99,248)(100,247)(101,246)(102,245)(103,244)(104,243)
(105,242)(106,241)(107,240)(108,239)(109,238)(110,237)(111,236)(112,235)
(113,234)(114,233)(115,232)(116,231)(117,230)(118,229)(119,228)(120,227)
(121,226)(122,225)(123,224)(124,223)(125,222)(126,221)(127,220)(128,219)
(129,218);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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(258)!( 44, 87)( 45, 88)( 46, 89)( 47, 90)( 48, 91)( 49, 92)( 50, 93)
( 51, 94)( 52, 95)( 53, 96)( 54, 97)( 55, 98)( 56, 99)( 57,100)( 58,101)
( 59,102)( 60,103)( 61,104)( 62,105)( 63,106)( 64,107)( 65,108)( 66,109)
( 67,110)( 68,111)( 69,112)( 70,113)( 71,114)( 72,115)( 73,116)( 74,117)
( 75,118)( 76,119)( 77,120)( 78,121)( 79,122)( 80,123)( 81,124)( 82,125)
( 83,126)( 84,127)( 85,128)( 86,129)(173,216)(174,217)(175,218)(176,219)
(177,220)(178,221)(179,222)(180,223)(181,224)(182,225)(183,226)(184,227)
(185,228)(186,229)(187,230)(188,231)(189,232)(190,233)(191,234)(192,235)
(193,236)(194,237)(195,238)(196,239)(197,240)(198,241)(199,242)(200,243)
(201,244)(202,245)(203,246)(204,247)(205,248)(206,249)(207,250)(208,251)
(209,252)(210,253)(211,254)(212,255)(213,256)(214,257)(215,258);
s1 := Sym(258)!(  1, 44)(  2, 86)(  3, 85)(  4, 84)(  5, 83)(  6, 82)(  7, 81)
(  8, 80)(  9, 79)( 10, 78)( 11, 77)( 12, 76)( 13, 75)( 14, 74)( 15, 73)
( 16, 72)( 17, 71)( 18, 70)( 19, 69)( 20, 68)( 21, 67)( 22, 66)( 23, 65)
( 24, 64)( 25, 63)( 26, 62)( 27, 61)( 28, 60)( 29, 59)( 30, 58)( 31, 57)
( 32, 56)( 33, 55)( 34, 54)( 35, 53)( 36, 52)( 37, 51)( 38, 50)( 39, 49)
( 40, 48)( 41, 47)( 42, 46)( 43, 45)( 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)(130,173)(131,215)(132,214)(133,213)(134,212)(135,211)(136,210)
(137,209)(138,208)(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)
(145,201)(146,200)(147,199)(148,198)(149,197)(150,196)(151,195)(152,194)
(153,193)(154,192)(155,191)(156,190)(157,189)(158,188)(159,187)(160,186)
(161,185)(162,184)(163,183)(164,182)(165,181)(166,180)(167,179)(168,178)
(169,177)(170,176)(171,175)(172,174)(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);
s2 := Sym(258)!(  1,131)(  2,130)(  3,172)(  4,171)(  5,170)(  6,169)(  7,168)
(  8,167)(  9,166)( 10,165)( 11,164)( 12,163)( 13,162)( 14,161)( 15,160)
( 16,159)( 17,158)( 18,157)( 19,156)( 20,155)( 21,154)( 22,153)( 23,152)
( 24,151)( 25,150)( 26,149)( 27,148)( 28,147)( 29,146)( 30,145)( 31,144)
( 32,143)( 33,142)( 34,141)( 35,140)( 36,139)( 37,138)( 38,137)( 39,136)
( 40,135)( 41,134)( 42,133)( 43,132)( 44,174)( 45,173)( 46,215)( 47,214)
( 48,213)( 49,212)( 50,211)( 51,210)( 52,209)( 53,208)( 54,207)( 55,206)
( 56,205)( 57,204)( 58,203)( 59,202)( 60,201)( 61,200)( 62,199)( 63,198)
( 64,197)( 65,196)( 66,195)( 67,194)( 68,193)( 69,192)( 70,191)( 71,190)
( 72,189)( 73,188)( 74,187)( 75,186)( 76,185)( 77,184)( 78,183)( 79,182)
( 80,181)( 81,180)( 82,179)( 83,178)( 84,177)( 85,176)( 86,175)( 87,217)
( 88,216)( 89,258)( 90,257)( 91,256)( 92,255)( 93,254)( 94,253)( 95,252)
( 96,251)( 97,250)( 98,249)( 99,248)(100,247)(101,246)(102,245)(103,244)
(104,243)(105,242)(106,241)(107,240)(108,239)(109,238)(110,237)(111,236)
(112,235)(113,234)(114,233)(115,232)(116,231)(117,230)(118,229)(119,228)
(120,227)(121,226)(122,225)(123,224)(124,223)(125,222)(126,221)(127,220)
(128,219)(129,218);
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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.
to this polytope