Polytope of Type {10,56}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,56}*1120
Also Known As : {10,56|2}. if this polytope has another name.
Group : SmallGroup(1120,307)
Rank : 3
Schlafli Type : {10,56}
Number of vertices, edges, etc : 10, 280, 56
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 : {10,28}*560
   4-fold quotients : {10,14}*280
   5-fold quotients : {2,56}*224
   7-fold quotients : {10,8}*160
   10-fold quotients : {2,28}*112
   14-fold quotients : {10,4}*80
   20-fold quotients : {2,14}*56
   28-fold quotients : {10,2}*40
   35-fold quotients : {2,8}*32
   40-fold quotients : {2,7}*28
   56-fold quotients : {5,2}*20
   70-fold quotients : {2,4}*16
   140-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope