Polytope of Type {8,70}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,70}*1120
Also Known As : {8,70|2}. if this polytope has another name.
Group : SmallGroup(1120,846)
Rank : 3
Schlafli Type : {8,70}
Number of vertices, edges, etc : 8, 280, 70
Order of s₀s₁s₂ : 280
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) :
   2-fold quotients : {4,70}*560
   4-fold quotients : {2,70}*280
   5-fold quotients : {8,14}*224
   7-fold quotients : {8,10}*160
   8-fold quotients : {2,35}*140
   10-fold quotients : {4,14}*112
   14-fold quotients : {4,10}*80
   20-fold quotients : {2,14}*56
   28-fold quotients : {2,10}*40
   35-fold quotients : {8,2}*32
   40-fold quotients : {2,7}*28
   56-fold quotients : {2,5}*20
   70-fold quotients : {4,2}*16
   140-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 71,106)( 72,107)( 73,108)( 74,109)( 75,110)( 76,111)( 77,112)( 78,113)
( 79,114)( 80,115)( 81,116)( 82,117)( 83,118)( 84,119)( 85,120)( 86,121)
( 87,122)( 88,123)( 89,124)( 90,125)( 91,126)( 92,127)( 93,128)( 94,129)
( 95,130)( 96,131)( 97,132)( 98,133)( 99,134)(100,135)(101,136)(102,137)
(103,138)(104,139)(105,140)(141,211)(142,212)(143,213)(144,214)(145,215)
(146,216)(147,217)(148,218)(149,219)(150,220)(151,221)(152,222)(153,223)
(154,224)(155,225)(156,226)(157,227)(158,228)(159,229)(160,230)(161,231)
(162,232)(163,233)(164,234)(165,235)(166,236)(167,237)(168,238)(169,239)
(170,240)(171,241)(172,242)(173,243)(174,244)(175,245)(176,246)(177,247)
(178,248)(179,249)(180,250)(181,251)(182,252)(183,253)(184,254)(185,255)
(186,256)(187,257)(188,258)(189,259)(190,260)(191,261)(192,262)(193,263)
(194,264)(195,265)(196,266)(197,267)(198,268)(199,269)(200,270)(201,271)
(202,272)(203,273)(204,274)(205,275)(206,276)(207,277)(208,278)(209,279)
(210,280);;
s1 := (  1,141)(  2,147)(  3,146)(  4,145)(  5,144)(  6,143)(  7,142)(  8,169)
(  9,175)( 10,174)( 11,173)( 12,172)( 13,171)( 14,170)( 15,162)( 16,168)
( 17,167)( 18,166)( 19,165)( 20,164)( 21,163)( 22,155)( 23,161)( 24,160)
( 25,159)( 26,158)( 27,157)( 28,156)( 29,148)( 30,154)( 31,153)( 32,152)
( 33,151)( 34,150)( 35,149)( 36,176)( 37,182)( 38,181)( 39,180)( 40,179)
( 41,178)( 42,177)( 43,204)( 44,210)( 45,209)( 46,208)( 47,207)( 48,206)
( 49,205)( 50,197)( 51,203)( 52,202)( 53,201)( 54,200)( 55,199)( 56,198)
( 57,190)( 58,196)( 59,195)( 60,194)( 61,193)( 62,192)( 63,191)( 64,183)
( 65,189)( 66,188)( 67,187)( 68,186)( 69,185)( 70,184)( 71,246)( 72,252)
( 73,251)( 74,250)( 75,249)( 76,248)( 77,247)( 78,274)( 79,280)( 80,279)
( 81,278)( 82,277)( 83,276)( 84,275)( 85,267)( 86,273)( 87,272)( 88,271)
( 89,270)( 90,269)( 91,268)( 92,260)( 93,266)( 94,265)( 95,264)( 96,263)
( 97,262)( 98,261)( 99,253)(100,259)(101,258)(102,257)(103,256)(104,255)
(105,254)(106,211)(107,217)(108,216)(109,215)(110,214)(111,213)(112,212)
(113,239)(114,245)(115,244)(116,243)(117,242)(118,241)(119,240)(120,232)
(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,225)(128,231)
(129,230)(130,229)(131,228)(132,227)(133,226)(134,218)(135,224)(136,223)
(137,222)(138,221)(139,220)(140,219);;
s2 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 30)
( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)( 24, 28)
( 25, 27)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)( 41, 46)( 42, 45)
( 50, 65)( 51, 64)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)( 57, 58)
( 59, 63)( 60, 62)( 71, 79)( 72, 78)( 73, 84)( 74, 83)( 75, 82)( 76, 81)
( 77, 80)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)
( 92, 93)( 94, 98)( 95, 97)(106,114)(107,113)(108,119)(109,118)(110,117)
(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)(124,138)(125,137)
(126,136)(127,128)(129,133)(130,132)(141,149)(142,148)(143,154)(144,153)
(145,152)(146,151)(147,150)(155,170)(156,169)(157,175)(158,174)(159,173)
(160,172)(161,171)(162,163)(164,168)(165,167)(176,184)(177,183)(178,189)
(179,188)(180,187)(181,186)(182,185)(190,205)(191,204)(192,210)(193,209)
(194,208)(195,207)(196,206)(197,198)(199,203)(200,202)(211,219)(212,218)
(213,224)(214,223)(215,222)(216,221)(217,220)(225,240)(226,239)(227,245)
(228,244)(229,243)(230,242)(231,241)(232,233)(234,238)(235,237)(246,254)
(247,253)(248,259)(249,258)(250,257)(251,256)(252,255)(260,275)(261,274)
(262,280)(263,279)(264,278)(265,277)(266,276)(267,268)(269,273)(270,272);;
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*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(280)!( 71,106)( 72,107)( 73,108)( 74,109)( 75,110)( 76,111)( 77,112)
( 78,113)( 79,114)( 80,115)( 81,116)( 82,117)( 83,118)( 84,119)( 85,120)
( 86,121)( 87,122)( 88,123)( 89,124)( 90,125)( 91,126)( 92,127)( 93,128)
( 94,129)( 95,130)( 96,131)( 97,132)( 98,133)( 99,134)(100,135)(101,136)
(102,137)(103,138)(104,139)(105,140)(141,211)(142,212)(143,213)(144,214)
(145,215)(146,216)(147,217)(148,218)(149,219)(150,220)(151,221)(152,222)
(153,223)(154,224)(155,225)(156,226)(157,227)(158,228)(159,229)(160,230)
(161,231)(162,232)(163,233)(164,234)(165,235)(166,236)(167,237)(168,238)
(169,239)(170,240)(171,241)(172,242)(173,243)(174,244)(175,245)(176,246)
(177,247)(178,248)(179,249)(180,250)(181,251)(182,252)(183,253)(184,254)
(185,255)(186,256)(187,257)(188,258)(189,259)(190,260)(191,261)(192,262)
(193,263)(194,264)(195,265)(196,266)(197,267)(198,268)(199,269)(200,270)
(201,271)(202,272)(203,273)(204,274)(205,275)(206,276)(207,277)(208,278)
(209,279)(210,280);
s1 := Sym(280)!(  1,141)(  2,147)(  3,146)(  4,145)(  5,144)(  6,143)(  7,142)
(  8,169)(  9,175)( 10,174)( 11,173)( 12,172)( 13,171)( 14,170)( 15,162)
( 16,168)( 17,167)( 18,166)( 19,165)( 20,164)( 21,163)( 22,155)( 23,161)
( 24,160)( 25,159)( 26,158)( 27,157)( 28,156)( 29,148)( 30,154)( 31,153)
( 32,152)( 33,151)( 34,150)( 35,149)( 36,176)( 37,182)( 38,181)( 39,180)
( 40,179)( 41,178)( 42,177)( 43,204)( 44,210)( 45,209)( 46,208)( 47,207)
( 48,206)( 49,205)( 50,197)( 51,203)( 52,202)( 53,201)( 54,200)( 55,199)
( 56,198)( 57,190)( 58,196)( 59,195)( 60,194)( 61,193)( 62,192)( 63,191)
( 64,183)( 65,189)( 66,188)( 67,187)( 68,186)( 69,185)( 70,184)( 71,246)
( 72,252)( 73,251)( 74,250)( 75,249)( 76,248)( 77,247)( 78,274)( 79,280)
( 80,279)( 81,278)( 82,277)( 83,276)( 84,275)( 85,267)( 86,273)( 87,272)
( 88,271)( 89,270)( 90,269)( 91,268)( 92,260)( 93,266)( 94,265)( 95,264)
( 96,263)( 97,262)( 98,261)( 99,253)(100,259)(101,258)(102,257)(103,256)
(104,255)(105,254)(106,211)(107,217)(108,216)(109,215)(110,214)(111,213)
(112,212)(113,239)(114,245)(115,244)(116,243)(117,242)(118,241)(119,240)
(120,232)(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,225)
(128,231)(129,230)(130,229)(131,228)(132,227)(133,226)(134,218)(135,224)
(136,223)(137,222)(138,221)(139,220)(140,219);
s2 := Sym(280)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)
( 15, 30)( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)
( 24, 28)( 25, 27)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)( 41, 46)
( 42, 45)( 50, 65)( 51, 64)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)
( 57, 58)( 59, 63)( 60, 62)( 71, 79)( 72, 78)( 73, 84)( 74, 83)( 75, 82)
( 76, 81)( 77, 80)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)
( 91,101)( 92, 93)( 94, 98)( 95, 97)(106,114)(107,113)(108,119)(109,118)
(110,117)(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)(124,138)
(125,137)(126,136)(127,128)(129,133)(130,132)(141,149)(142,148)(143,154)
(144,153)(145,152)(146,151)(147,150)(155,170)(156,169)(157,175)(158,174)
(159,173)(160,172)(161,171)(162,163)(164,168)(165,167)(176,184)(177,183)
(178,189)(179,188)(180,187)(181,186)(182,185)(190,205)(191,204)(192,210)
(193,209)(194,208)(195,207)(196,206)(197,198)(199,203)(200,202)(211,219)
(212,218)(213,224)(214,223)(215,222)(216,221)(217,220)(225,240)(226,239)
(227,245)(228,244)(229,243)(230,242)(231,241)(232,233)(234,238)(235,237)
(246,254)(247,253)(248,259)(249,258)(250,257)(251,256)(252,255)(260,275)
(261,274)(262,280)(263,279)(264,278)(265,277)(266,276)(267,268)(269,273)
(270,272);
poly := sub<Sym(280)|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*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 >;

References : None.
to this polytope