Polytope of Type {56,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {56,10}*1120
Also Known As : {56,10|2}. if this polytope has another name.
Group : SmallGroup(1120,307)
Rank : 3
Schlafli Type : {56,10}
Number of vertices, edges, etc : 56, 280, 10
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 : {28,10}*560
   4-fold quotients : {14,10}*280
   5-fold quotients : {56,2}*224
   7-fold quotients : {8,10}*160
   10-fold quotients : {28,2}*112
   14-fold quotients : {4,10}*80
   20-fold quotients : {14,2}*56
   28-fold quotients : {2,10}*40
   35-fold quotients : {8,2}*32
   40-fold quotients : {7,2}*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 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)
( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 71,106)( 72,112)
( 73,111)( 74,110)( 75,109)( 76,108)( 77,107)( 78,113)( 79,119)( 80,118)
( 81,117)( 82,116)( 83,115)( 84,114)( 85,120)( 86,126)( 87,125)( 88,124)
( 89,123)( 90,122)( 91,121)( 92,127)( 93,133)( 94,132)( 95,131)( 96,130)
( 97,129)( 98,128)( 99,134)(100,140)(101,139)(102,138)(103,137)(104,136)
(105,135)(141,211)(142,217)(143,216)(144,215)(145,214)(146,213)(147,212)
(148,218)(149,224)(150,223)(151,222)(152,221)(153,220)(154,219)(155,225)
(156,231)(157,230)(158,229)(159,228)(160,227)(161,226)(162,232)(163,238)
(164,237)(165,236)(166,235)(167,234)(168,233)(169,239)(170,245)(171,244)
(172,243)(173,242)(174,241)(175,240)(176,246)(177,252)(178,251)(179,250)
(180,249)(181,248)(182,247)(183,253)(184,259)(185,258)(186,257)(187,256)
(188,255)(189,254)(190,260)(191,266)(192,265)(193,264)(194,263)(195,262)
(196,261)(197,267)(198,273)(199,272)(200,271)(201,270)(202,269)(203,268)
(204,274)(205,280)(206,279)(207,278)(208,277)(209,276)(210,275);;
s1 := (  1,142)(  2,141)(  3,147)(  4,146)(  5,145)(  6,144)(  7,143)(  8,170)
(  9,169)( 10,175)( 11,174)( 12,173)( 13,172)( 14,171)( 15,163)( 16,162)
( 17,168)( 18,167)( 19,166)( 20,165)( 21,164)( 22,156)( 23,155)( 24,161)
( 25,160)( 26,159)( 27,158)( 28,157)( 29,149)( 30,148)( 31,154)( 32,153)
( 33,152)( 34,151)( 35,150)( 36,177)( 37,176)( 38,182)( 39,181)( 40,180)
( 41,179)( 42,178)( 43,205)( 44,204)( 45,210)( 46,209)( 47,208)( 48,207)
( 49,206)( 50,198)( 51,197)( 52,203)( 53,202)( 54,201)( 55,200)( 56,199)
( 57,191)( 58,190)( 59,196)( 60,195)( 61,194)( 62,193)( 63,192)( 64,184)
( 65,183)( 66,189)( 67,188)( 68,187)( 69,186)( 70,185)( 71,247)( 72,246)
( 73,252)( 74,251)( 75,250)( 76,249)( 77,248)( 78,275)( 79,274)( 80,280)
( 81,279)( 82,278)( 83,277)( 84,276)( 85,268)( 86,267)( 87,273)( 88,272)
( 89,271)( 90,270)( 91,269)( 92,261)( 93,260)( 94,266)( 95,265)( 96,264)
( 97,263)( 98,262)( 99,254)(100,253)(101,259)(102,258)(103,257)(104,256)
(105,255)(106,212)(107,211)(108,217)(109,216)(110,215)(111,214)(112,213)
(113,240)(114,239)(115,245)(116,244)(117,243)(118,242)(119,241)(120,233)
(121,232)(122,238)(123,237)(124,236)(125,235)(126,234)(127,226)(128,225)
(129,231)(130,230)(131,229)(132,228)(133,227)(134,219)(135,218)(136,224)
(137,223)(138,222)(139,221)(140,220);;
s2 := (  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)( 15, 29)
( 16, 30)( 17, 31)( 18, 32)( 19, 33)( 20, 34)( 21, 35)( 36, 43)( 37, 44)
( 38, 45)( 39, 46)( 40, 47)( 41, 48)( 42, 49)( 50, 64)( 51, 65)( 52, 66)
( 53, 67)( 54, 68)( 55, 69)( 56, 70)( 71, 78)( 72, 79)( 73, 80)( 74, 81)
( 75, 82)( 76, 83)( 77, 84)( 85, 99)( 86,100)( 87,101)( 88,102)( 89,103)
( 90,104)( 91,105)(106,113)(107,114)(108,115)(109,116)(110,117)(111,118)
(112,119)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)
(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,169)
(156,170)(157,171)(158,172)(159,173)(160,174)(161,175)(176,183)(177,184)
(178,185)(179,186)(180,187)(181,188)(182,189)(190,204)(191,205)(192,206)
(193,207)(194,208)(195,209)(196,210)(211,218)(212,219)(213,220)(214,221)
(215,222)(216,223)(217,224)(225,239)(226,240)(227,241)(228,242)(229,243)
(230,244)(231,245)(246,253)(247,254)(248,255)(249,256)(250,257)(251,258)
(252,259)(260,274)(261,275)(262,276)(263,277)(264,278)(265,279)(266,280);;
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*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope