Polytope of Type {28,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,20}*1120
Also Known As : {28,20|2}. if this polytope has another name.
Group : SmallGroup(1120,530)
Rank : 3
Schlafli Type : {28,20}
Number of vertices, edges, etc : 28, 280, 20
Order of s₀s₁s₂ : 140
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 : {14,20}*560, {28,10}*560
   4-fold quotients : {14,10}*280
   5-fold quotients : {28,4}*224
   7-fold quotients : {4,20}*160
   10-fold quotients : {28,2}*112, {14,4}*112
   14-fold quotients : {2,20}*80, {4,10}*80
   20-fold quotients : {14,2}*56
   28-fold quotients : {2,10}*40
   35-fold quotients : {4,4}*32
   40-fold quotients : {7,2}*28
   56-fold quotients : {2,5}*20
   70-fold quotients : {2,4}*16, {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)( 72, 77)( 73, 76)
( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)
( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)
(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)(129,132)
(130,131)(135,140)(136,139)(137,138)(141,176)(142,182)(143,181)(144,180)
(145,179)(146,178)(147,177)(148,183)(149,189)(150,188)(151,187)(152,186)
(153,185)(154,184)(155,190)(156,196)(157,195)(158,194)(159,193)(160,192)
(161,191)(162,197)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198)
(169,204)(170,210)(171,209)(172,208)(173,207)(174,206)(175,205)(211,246)
(212,252)(213,251)(214,250)(215,249)(216,248)(217,247)(218,253)(219,259)
(220,258)(221,257)(222,256)(223,255)(224,254)(225,260)(226,266)(227,265)
(228,264)(229,263)(230,262)(231,261)(232,267)(233,273)(234,272)(235,271)
(236,270)(237,269)(238,268)(239,274)(240,280)(241,279)(242,278)(243,277)
(244,276)(245,275);;
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)( 71,212)( 72,211)
( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,240)( 79,239)( 80,245)
( 81,244)( 82,243)( 83,242)( 84,241)( 85,233)( 86,232)( 87,238)( 88,237)
( 89,236)( 90,235)( 91,234)( 92,226)( 93,225)( 94,231)( 95,230)( 96,229)
( 97,228)( 98,227)( 99,219)(100,218)(101,224)(102,223)(103,222)(104,221)
(105,220)(106,247)(107,246)(108,252)(109,251)(110,250)(111,249)(112,248)
(113,275)(114,274)(115,280)(116,279)(117,278)(118,277)(119,276)(120,268)
(121,267)(122,273)(123,272)(124,271)(125,270)(126,269)(127,261)(128,260)
(129,266)(130,265)(131,264)(132,263)(133,262)(134,254)(135,253)(136,259)
(137,258)(138,257)(139,256)(140,255);;
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)
(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,211)
(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,239)(156,240)
(157,241)(158,242)(159,243)(160,244)(161,245)(162,232)(163,233)(164,234)
(165,235)(166,236)(167,237)(168,238)(169,225)(170,226)(171,227)(172,228)
(173,229)(174,230)(175,231)(176,253)(177,254)(178,255)(179,256)(180,257)
(181,258)(182,259)(183,246)(184,247)(185,248)(186,249)(187,250)(188,251)
(189,252)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)
(197,267)(198,268)(199,269)(200,270)(201,271)(202,272)(203,273)(204,260)
(205,261)(206,262)(207,263)(208,264)(209,265)(210,266);;
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*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 ];;
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)( 72, 77)
( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)
( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)
(109,110)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)
(129,132)(130,131)(135,140)(136,139)(137,138)(141,176)(142,182)(143,181)
(144,180)(145,179)(146,178)(147,177)(148,183)(149,189)(150,188)(151,187)
(152,186)(153,185)(154,184)(155,190)(156,196)(157,195)(158,194)(159,193)
(160,192)(161,191)(162,197)(163,203)(164,202)(165,201)(166,200)(167,199)
(168,198)(169,204)(170,210)(171,209)(172,208)(173,207)(174,206)(175,205)
(211,246)(212,252)(213,251)(214,250)(215,249)(216,248)(217,247)(218,253)
(219,259)(220,258)(221,257)(222,256)(223,255)(224,254)(225,260)(226,266)
(227,265)(228,264)(229,263)(230,262)(231,261)(232,267)(233,273)(234,272)
(235,271)(236,270)(237,269)(238,268)(239,274)(240,280)(241,279)(242,278)
(243,277)(244,276)(245,275);
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)( 71,212)
( 72,211)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,240)( 79,239)
( 80,245)( 81,244)( 82,243)( 83,242)( 84,241)( 85,233)( 86,232)( 87,238)
( 88,237)( 89,236)( 90,235)( 91,234)( 92,226)( 93,225)( 94,231)( 95,230)
( 96,229)( 97,228)( 98,227)( 99,219)(100,218)(101,224)(102,223)(103,222)
(104,221)(105,220)(106,247)(107,246)(108,252)(109,251)(110,250)(111,249)
(112,248)(113,275)(114,274)(115,280)(116,279)(117,278)(118,277)(119,276)
(120,268)(121,267)(122,273)(123,272)(124,271)(125,270)(126,269)(127,261)
(128,260)(129,266)(130,265)(131,264)(132,263)(133,262)(134,254)(135,253)
(136,259)(137,258)(138,257)(139,256)(140,255);
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)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)
(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,239)
(156,240)(157,241)(158,242)(159,243)(160,244)(161,245)(162,232)(163,233)
(164,234)(165,235)(166,236)(167,237)(168,238)(169,225)(170,226)(171,227)
(172,228)(173,229)(174,230)(175,231)(176,253)(177,254)(178,255)(179,256)
(180,257)(181,258)(182,259)(183,246)(184,247)(185,248)(186,249)(187,250)
(188,251)(189,252)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)
(196,280)(197,267)(198,268)(199,269)(200,270)(201,271)(202,272)(203,273)
(204,260)(205,261)(206,262)(207,263)(208,264)(209,265)(210,266);
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*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 >;

References : None.
to this polytope