Polytope of Type {4,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,10}*640a
if this polytope has a name.
Group : SmallGroup(640,21465)
Rank : 3
Schlafli Type : {4,10}
Number of vertices, edges, etc : 32, 160, 80
Order of s₀s₁s₂ : 20
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,10,2} of size 1280
Vertex Figure Of :
   {2,4,10} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,5}*320
   4-fold quotients : {4,5}*160
   32-fold quotients : {2,5}*20
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,10}*1280a, {8,10}*1280b, {4,10}*1280b
   3-fold covers : {4,30}*1920c
Permutation Representation (GAP) :

s0 := (  1,177)(  2,178)(  3,180)(  4,179)(  5,182)(  6,181)(  7,183)(  8,184)
(  9,186)( 10,185)( 11,187)( 12,188)( 13,189)( 14,190)( 15,192)( 16,191)
( 17,162)( 18,161)( 19,163)( 20,164)( 21,165)( 22,166)( 23,168)( 24,167)
( 25,169)( 26,170)( 27,172)( 28,171)( 29,174)( 30,173)( 31,175)( 32,176)
( 33,209)( 34,210)( 35,212)( 36,211)( 37,214)( 38,213)( 39,215)( 40,216)
( 41,218)( 42,217)( 43,219)( 44,220)( 45,221)( 46,222)( 47,224)( 48,223)
( 49,194)( 50,193)( 51,195)( 52,196)( 53,197)( 54,198)( 55,200)( 56,199)
( 57,201)( 58,202)( 59,204)( 60,203)( 61,206)( 62,205)( 63,207)( 64,208)
( 65,241)( 66,242)( 67,244)( 68,243)( 69,246)( 70,245)( 71,247)( 72,248)
( 73,250)( 74,249)( 75,251)( 76,252)( 77,253)( 78,254)( 79,256)( 80,255)
( 81,226)( 82,225)( 83,227)( 84,228)( 85,229)( 86,230)( 87,232)( 88,231)
( 89,233)( 90,234)( 91,236)( 92,235)( 93,238)( 94,237)( 95,239)( 96,240)
( 97,273)( 98,274)( 99,276)(100,275)(101,278)(102,277)(103,279)(104,280)
(105,282)(106,281)(107,283)(108,284)(109,285)(110,286)(111,288)(112,287)
(113,258)(114,257)(115,259)(116,260)(117,261)(118,262)(119,264)(120,263)
(121,265)(122,266)(123,268)(124,267)(125,270)(126,269)(127,271)(128,272)
(129,305)(130,306)(131,308)(132,307)(133,310)(134,309)(135,311)(136,312)
(137,314)(138,313)(139,315)(140,316)(141,317)(142,318)(143,320)(144,319)
(145,290)(146,289)(147,291)(148,292)(149,293)(150,294)(151,296)(152,295)
(153,297)(154,298)(155,300)(156,299)(157,302)(158,301)(159,303)(160,304);;
s1 := (  1, 33)(  2, 34)(  3, 55)(  4, 56)(  5, 57)(  6, 58)(  7, 47)(  8, 48)
(  9, 42)( 10, 41)( 11, 63)( 12, 64)( 13, 50)( 14, 49)( 15, 39)( 16, 40)
( 17, 46)( 18, 45)( 19, 59)( 20, 60)( 21, 54)( 22, 53)( 23, 35)( 24, 36)
( 25, 37)( 26, 38)( 27, 51)( 28, 52)( 29, 61)( 30, 62)( 31, 43)( 32, 44)
( 65,129)( 66,130)( 67,151)( 68,152)( 69,153)( 70,154)( 71,143)( 72,144)
( 73,138)( 74,137)( 75,159)( 76,160)( 77,146)( 78,145)( 79,135)( 80,136)
( 81,142)( 82,141)( 83,155)( 84,156)( 85,150)( 86,149)( 87,131)( 88,132)
( 89,133)( 90,134)( 91,147)( 92,148)( 93,157)( 94,158)( 95,139)( 96,140)
( 99,119)(100,120)(101,121)(102,122)(103,111)(104,112)(105,106)(107,127)
(108,128)(109,114)(110,113)(115,123)(116,124)(117,118)(161,194)(162,193)
(163,216)(164,215)(165,218)(166,217)(167,208)(168,207)(169,201)(170,202)
(171,224)(172,223)(173,209)(174,210)(175,200)(176,199)(177,205)(178,206)
(179,220)(180,219)(181,213)(182,214)(183,196)(184,195)(185,198)(186,197)
(187,212)(188,211)(189,222)(190,221)(191,204)(192,203)(225,290)(226,289)
(227,312)(228,311)(229,314)(230,313)(231,304)(232,303)(233,297)(234,298)
(235,320)(236,319)(237,305)(238,306)(239,296)(240,295)(241,301)(242,302)
(243,316)(244,315)(245,309)(246,310)(247,292)(248,291)(249,294)(250,293)
(251,308)(252,307)(253,318)(254,317)(255,300)(256,299)(257,258)(259,280)
(260,279)(261,282)(262,281)(263,272)(264,271)(267,288)(268,287)(269,273)
(270,274)(275,284)(276,283)(285,286);;
s2 := (  1,130)(  2,129)(  3,147)(  4,148)(  5,150)(  6,149)(  7,135)(  8,136)
(  9,159)( 10,160)( 11,142)( 12,141)( 13,140)( 14,139)( 15,153)( 16,154)
( 17,145)( 18,146)( 19,131)( 20,132)( 21,134)( 22,133)( 23,152)( 24,151)
( 25,143)( 26,144)( 27,157)( 28,158)( 29,155)( 30,156)( 31,137)( 32,138)
( 33, 98)( 34, 97)( 35,115)( 36,116)( 37,118)( 38,117)( 39,103)( 40,104)
( 41,127)( 42,128)( 43,110)( 44,109)( 45,108)( 46,107)( 47,121)( 48,122)
( 49,113)( 50,114)( 51, 99)( 52,100)( 53,102)( 54,101)( 55,120)( 56,119)
( 57,111)( 58,112)( 59,125)( 60,126)( 61,123)( 62,124)( 63,105)( 64,106)
( 65, 66)( 67, 83)( 68, 84)( 69, 86)( 70, 85)( 73, 95)( 74, 96)( 75, 78)
( 76, 77)( 79, 89)( 80, 90)( 87, 88)( 91, 93)( 92, 94)(161,289)(162,290)
(163,308)(164,307)(165,309)(166,310)(167,296)(168,295)(169,320)(170,319)
(171,301)(172,302)(173,299)(174,300)(175,314)(176,313)(177,306)(178,305)
(179,292)(180,291)(181,293)(182,294)(183,311)(184,312)(185,304)(186,303)
(187,318)(188,317)(189,316)(190,315)(191,298)(192,297)(193,257)(194,258)
(195,276)(196,275)(197,277)(198,278)(199,264)(200,263)(201,288)(202,287)
(203,269)(204,270)(205,267)(206,268)(207,282)(208,281)(209,274)(210,273)
(211,260)(212,259)(213,261)(214,262)(215,279)(216,280)(217,272)(218,271)
(219,286)(220,285)(221,284)(222,283)(223,266)(224,265)(227,244)(228,243)
(229,245)(230,246)(231,232)(233,256)(234,255)(235,237)(236,238)(239,250)
(240,249)(241,242)(251,254)(252,253);;
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*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(320)!(  1,177)(  2,178)(  3,180)(  4,179)(  5,182)(  6,181)(  7,183)
(  8,184)(  9,186)( 10,185)( 11,187)( 12,188)( 13,189)( 14,190)( 15,192)
( 16,191)( 17,162)( 18,161)( 19,163)( 20,164)( 21,165)( 22,166)( 23,168)
( 24,167)( 25,169)( 26,170)( 27,172)( 28,171)( 29,174)( 30,173)( 31,175)
( 32,176)( 33,209)( 34,210)( 35,212)( 36,211)( 37,214)( 38,213)( 39,215)
( 40,216)( 41,218)( 42,217)( 43,219)( 44,220)( 45,221)( 46,222)( 47,224)
( 48,223)( 49,194)( 50,193)( 51,195)( 52,196)( 53,197)( 54,198)( 55,200)
( 56,199)( 57,201)( 58,202)( 59,204)( 60,203)( 61,206)( 62,205)( 63,207)
( 64,208)( 65,241)( 66,242)( 67,244)( 68,243)( 69,246)( 70,245)( 71,247)
( 72,248)( 73,250)( 74,249)( 75,251)( 76,252)( 77,253)( 78,254)( 79,256)
( 80,255)( 81,226)( 82,225)( 83,227)( 84,228)( 85,229)( 86,230)( 87,232)
( 88,231)( 89,233)( 90,234)( 91,236)( 92,235)( 93,238)( 94,237)( 95,239)
( 96,240)( 97,273)( 98,274)( 99,276)(100,275)(101,278)(102,277)(103,279)
(104,280)(105,282)(106,281)(107,283)(108,284)(109,285)(110,286)(111,288)
(112,287)(113,258)(114,257)(115,259)(116,260)(117,261)(118,262)(119,264)
(120,263)(121,265)(122,266)(123,268)(124,267)(125,270)(126,269)(127,271)
(128,272)(129,305)(130,306)(131,308)(132,307)(133,310)(134,309)(135,311)
(136,312)(137,314)(138,313)(139,315)(140,316)(141,317)(142,318)(143,320)
(144,319)(145,290)(146,289)(147,291)(148,292)(149,293)(150,294)(151,296)
(152,295)(153,297)(154,298)(155,300)(156,299)(157,302)(158,301)(159,303)
(160,304);
s1 := Sym(320)!(  1, 33)(  2, 34)(  3, 55)(  4, 56)(  5, 57)(  6, 58)(  7, 47)
(  8, 48)(  9, 42)( 10, 41)( 11, 63)( 12, 64)( 13, 50)( 14, 49)( 15, 39)
( 16, 40)( 17, 46)( 18, 45)( 19, 59)( 20, 60)( 21, 54)( 22, 53)( 23, 35)
( 24, 36)( 25, 37)( 26, 38)( 27, 51)( 28, 52)( 29, 61)( 30, 62)( 31, 43)
( 32, 44)( 65,129)( 66,130)( 67,151)( 68,152)( 69,153)( 70,154)( 71,143)
( 72,144)( 73,138)( 74,137)( 75,159)( 76,160)( 77,146)( 78,145)( 79,135)
( 80,136)( 81,142)( 82,141)( 83,155)( 84,156)( 85,150)( 86,149)( 87,131)
( 88,132)( 89,133)( 90,134)( 91,147)( 92,148)( 93,157)( 94,158)( 95,139)
( 96,140)( 99,119)(100,120)(101,121)(102,122)(103,111)(104,112)(105,106)
(107,127)(108,128)(109,114)(110,113)(115,123)(116,124)(117,118)(161,194)
(162,193)(163,216)(164,215)(165,218)(166,217)(167,208)(168,207)(169,201)
(170,202)(171,224)(172,223)(173,209)(174,210)(175,200)(176,199)(177,205)
(178,206)(179,220)(180,219)(181,213)(182,214)(183,196)(184,195)(185,198)
(186,197)(187,212)(188,211)(189,222)(190,221)(191,204)(192,203)(225,290)
(226,289)(227,312)(228,311)(229,314)(230,313)(231,304)(232,303)(233,297)
(234,298)(235,320)(236,319)(237,305)(238,306)(239,296)(240,295)(241,301)
(242,302)(243,316)(244,315)(245,309)(246,310)(247,292)(248,291)(249,294)
(250,293)(251,308)(252,307)(253,318)(254,317)(255,300)(256,299)(257,258)
(259,280)(260,279)(261,282)(262,281)(263,272)(264,271)(267,288)(268,287)
(269,273)(270,274)(275,284)(276,283)(285,286);
s2 := Sym(320)!(  1,130)(  2,129)(  3,147)(  4,148)(  5,150)(  6,149)(  7,135)
(  8,136)(  9,159)( 10,160)( 11,142)( 12,141)( 13,140)( 14,139)( 15,153)
( 16,154)( 17,145)( 18,146)( 19,131)( 20,132)( 21,134)( 22,133)( 23,152)
( 24,151)( 25,143)( 26,144)( 27,157)( 28,158)( 29,155)( 30,156)( 31,137)
( 32,138)( 33, 98)( 34, 97)( 35,115)( 36,116)( 37,118)( 38,117)( 39,103)
( 40,104)( 41,127)( 42,128)( 43,110)( 44,109)( 45,108)( 46,107)( 47,121)
( 48,122)( 49,113)( 50,114)( 51, 99)( 52,100)( 53,102)( 54,101)( 55,120)
( 56,119)( 57,111)( 58,112)( 59,125)( 60,126)( 61,123)( 62,124)( 63,105)
( 64,106)( 65, 66)( 67, 83)( 68, 84)( 69, 86)( 70, 85)( 73, 95)( 74, 96)
( 75, 78)( 76, 77)( 79, 89)( 80, 90)( 87, 88)( 91, 93)( 92, 94)(161,289)
(162,290)(163,308)(164,307)(165,309)(166,310)(167,296)(168,295)(169,320)
(170,319)(171,301)(172,302)(173,299)(174,300)(175,314)(176,313)(177,306)
(178,305)(179,292)(180,291)(181,293)(182,294)(183,311)(184,312)(185,304)
(186,303)(187,318)(188,317)(189,316)(190,315)(191,298)(192,297)(193,257)
(194,258)(195,276)(196,275)(197,277)(198,278)(199,264)(200,263)(201,288)
(202,287)(203,269)(204,270)(205,267)(206,268)(207,282)(208,281)(209,274)
(210,273)(211,260)(212,259)(213,261)(214,262)(215,279)(216,280)(217,272)
(218,271)(219,286)(220,285)(221,284)(222,283)(223,266)(224,265)(227,244)
(228,243)(229,245)(230,246)(231,232)(233,256)(234,255)(235,237)(236,238)
(239,250)(240,249)(241,242)(251,254)(252,253);
poly := sub<Sym(320)|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*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References : None.
to this polytope