Polytope of Type {2,4,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,10}*1280a
if this polytope has a name.
Group : SmallGroup(1280,1116454)
Rank : 4
Schlafli Type : {2,4,10}
Number of vertices, edges, etc : 2, 32, 160, 80
Order of s₀s₁s₂s₃ : 20
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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 : {2,4,5}*640
   4-fold quotients : {2,4,5}*320
   32-fold quotients : {2,2,5}*40
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  3,179)(  4,180)(  5,182)(  6,181)(  7,184)(  8,183)(  9,185)( 10,186)
( 11,188)( 12,187)( 13,189)( 14,190)( 15,191)( 16,192)( 17,194)( 18,193)
( 19,164)( 20,163)( 21,165)( 22,166)( 23,167)( 24,168)( 25,170)( 26,169)
( 27,171)( 28,172)( 29,174)( 30,173)( 31,176)( 32,175)( 33,177)( 34,178)
( 35,211)( 36,212)( 37,214)( 38,213)( 39,216)( 40,215)( 41,217)( 42,218)
( 43,220)( 44,219)( 45,221)( 46,222)( 47,223)( 48,224)( 49,226)( 50,225)
( 51,196)( 52,195)( 53,197)( 54,198)( 55,199)( 56,200)( 57,202)( 58,201)
( 59,203)( 60,204)( 61,206)( 62,205)( 63,208)( 64,207)( 65,209)( 66,210)
( 67,243)( 68,244)( 69,246)( 70,245)( 71,248)( 72,247)( 73,249)( 74,250)
( 75,252)( 76,251)( 77,253)( 78,254)( 79,255)( 80,256)( 81,258)( 82,257)
( 83,228)( 84,227)( 85,229)( 86,230)( 87,231)( 88,232)( 89,234)( 90,233)
( 91,235)( 92,236)( 93,238)( 94,237)( 95,240)( 96,239)( 97,241)( 98,242)
( 99,275)(100,276)(101,278)(102,277)(103,280)(104,279)(105,281)(106,282)
(107,284)(108,283)(109,285)(110,286)(111,287)(112,288)(113,290)(114,289)
(115,260)(116,259)(117,261)(118,262)(119,263)(120,264)(121,266)(122,265)
(123,267)(124,268)(125,270)(126,269)(127,272)(128,271)(129,273)(130,274)
(131,307)(132,308)(133,310)(134,309)(135,312)(136,311)(137,313)(138,314)
(139,316)(140,315)(141,317)(142,318)(143,319)(144,320)(145,322)(146,321)
(147,292)(148,291)(149,293)(150,294)(151,295)(152,296)(153,298)(154,297)
(155,299)(156,300)(157,302)(158,301)(159,304)(160,303)(161,305)(162,306);;
s2 := (  3, 35)(  4, 36)(  5, 57)(  6, 58)(  7, 59)(  8, 60)(  9, 49)( 10, 50)
( 11, 44)( 12, 43)( 13, 65)( 14, 66)( 15, 52)( 16, 51)( 17, 41)( 18, 42)
( 19, 48)( 20, 47)( 21, 61)( 22, 62)( 23, 56)( 24, 55)( 25, 37)( 26, 38)
( 27, 39)( 28, 40)( 29, 53)( 30, 54)( 31, 63)( 32, 64)( 33, 45)( 34, 46)
( 67,131)( 68,132)( 69,153)( 70,154)( 71,155)( 72,156)( 73,145)( 74,146)
( 75,140)( 76,139)( 77,161)( 78,162)( 79,148)( 80,147)( 81,137)( 82,138)
( 83,144)( 84,143)( 85,157)( 86,158)( 87,152)( 88,151)( 89,133)( 90,134)
( 91,135)( 92,136)( 93,149)( 94,150)( 95,159)( 96,160)( 97,141)( 98,142)
(101,121)(102,122)(103,123)(104,124)(105,113)(106,114)(107,108)(109,129)
(110,130)(111,116)(112,115)(117,125)(118,126)(119,120)(163,196)(164,195)
(165,218)(166,217)(167,220)(168,219)(169,210)(170,209)(171,203)(172,204)
(173,226)(174,225)(175,211)(176,212)(177,202)(178,201)(179,207)(180,208)
(181,222)(182,221)(183,215)(184,216)(185,198)(186,197)(187,200)(188,199)
(189,214)(190,213)(191,224)(192,223)(193,206)(194,205)(227,292)(228,291)
(229,314)(230,313)(231,316)(232,315)(233,306)(234,305)(235,299)(236,300)
(237,322)(238,321)(239,307)(240,308)(241,298)(242,297)(243,303)(244,304)
(245,318)(246,317)(247,311)(248,312)(249,294)(250,293)(251,296)(252,295)
(253,310)(254,309)(255,320)(256,319)(257,302)(258,301)(259,260)(261,282)
(262,281)(263,284)(264,283)(265,274)(266,273)(269,290)(270,289)(271,275)
(272,276)(277,286)(278,285)(287,288);;
s3 := (  3,132)(  4,131)(  5,149)(  6,150)(  7,152)(  8,151)(  9,137)( 10,138)
( 11,161)( 12,162)( 13,144)( 14,143)( 15,142)( 16,141)( 17,155)( 18,156)
( 19,147)( 20,148)( 21,133)( 22,134)( 23,136)( 24,135)( 25,154)( 26,153)
( 27,145)( 28,146)( 29,159)( 30,160)( 31,157)( 32,158)( 33,139)( 34,140)
( 35,100)( 36, 99)( 37,117)( 38,118)( 39,120)( 40,119)( 41,105)( 42,106)
( 43,129)( 44,130)( 45,112)( 46,111)( 47,110)( 48,109)( 49,123)( 50,124)
( 51,115)( 52,116)( 53,101)( 54,102)( 55,104)( 56,103)( 57,122)( 58,121)
( 59,113)( 60,114)( 61,127)( 62,128)( 63,125)( 64,126)( 65,107)( 66,108)
( 67, 68)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 75, 97)( 76, 98)( 77, 80)
( 78, 79)( 81, 91)( 82, 92)( 89, 90)( 93, 95)( 94, 96)(163,291)(164,292)
(165,310)(166,309)(167,311)(168,312)(169,298)(170,297)(171,322)(172,321)
(173,303)(174,304)(175,301)(176,302)(177,316)(178,315)(179,308)(180,307)
(181,294)(182,293)(183,295)(184,296)(185,313)(186,314)(187,306)(188,305)
(189,320)(190,319)(191,318)(192,317)(193,300)(194,299)(195,259)(196,260)
(197,278)(198,277)(199,279)(200,280)(201,266)(202,265)(203,290)(204,289)
(205,271)(206,272)(207,269)(208,270)(209,284)(210,283)(211,276)(212,275)
(213,262)(214,261)(215,263)(216,264)(217,281)(218,282)(219,274)(220,273)
(221,288)(222,287)(223,286)(224,285)(225,268)(226,267)(229,246)(230,245)
(231,247)(232,248)(233,234)(235,258)(236,257)(237,239)(238,240)(241,252)
(242,251)(243,244)(253,256)(254,255);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(322)!(1,2);
s1 := Sym(322)!(  3,179)(  4,180)(  5,182)(  6,181)(  7,184)(  8,183)(  9,185)
( 10,186)( 11,188)( 12,187)( 13,189)( 14,190)( 15,191)( 16,192)( 17,194)
( 18,193)( 19,164)( 20,163)( 21,165)( 22,166)( 23,167)( 24,168)( 25,170)
( 26,169)( 27,171)( 28,172)( 29,174)( 30,173)( 31,176)( 32,175)( 33,177)
( 34,178)( 35,211)( 36,212)( 37,214)( 38,213)( 39,216)( 40,215)( 41,217)
( 42,218)( 43,220)( 44,219)( 45,221)( 46,222)( 47,223)( 48,224)( 49,226)
( 50,225)( 51,196)( 52,195)( 53,197)( 54,198)( 55,199)( 56,200)( 57,202)
( 58,201)( 59,203)( 60,204)( 61,206)( 62,205)( 63,208)( 64,207)( 65,209)
( 66,210)( 67,243)( 68,244)( 69,246)( 70,245)( 71,248)( 72,247)( 73,249)
( 74,250)( 75,252)( 76,251)( 77,253)( 78,254)( 79,255)( 80,256)( 81,258)
( 82,257)( 83,228)( 84,227)( 85,229)( 86,230)( 87,231)( 88,232)( 89,234)
( 90,233)( 91,235)( 92,236)( 93,238)( 94,237)( 95,240)( 96,239)( 97,241)
( 98,242)( 99,275)(100,276)(101,278)(102,277)(103,280)(104,279)(105,281)
(106,282)(107,284)(108,283)(109,285)(110,286)(111,287)(112,288)(113,290)
(114,289)(115,260)(116,259)(117,261)(118,262)(119,263)(120,264)(121,266)
(122,265)(123,267)(124,268)(125,270)(126,269)(127,272)(128,271)(129,273)
(130,274)(131,307)(132,308)(133,310)(134,309)(135,312)(136,311)(137,313)
(138,314)(139,316)(140,315)(141,317)(142,318)(143,319)(144,320)(145,322)
(146,321)(147,292)(148,291)(149,293)(150,294)(151,295)(152,296)(153,298)
(154,297)(155,299)(156,300)(157,302)(158,301)(159,304)(160,303)(161,305)
(162,306);
s2 := Sym(322)!(  3, 35)(  4, 36)(  5, 57)(  6, 58)(  7, 59)(  8, 60)(  9, 49)
( 10, 50)( 11, 44)( 12, 43)( 13, 65)( 14, 66)( 15, 52)( 16, 51)( 17, 41)
( 18, 42)( 19, 48)( 20, 47)( 21, 61)( 22, 62)( 23, 56)( 24, 55)( 25, 37)
( 26, 38)( 27, 39)( 28, 40)( 29, 53)( 30, 54)( 31, 63)( 32, 64)( 33, 45)
( 34, 46)( 67,131)( 68,132)( 69,153)( 70,154)( 71,155)( 72,156)( 73,145)
( 74,146)( 75,140)( 76,139)( 77,161)( 78,162)( 79,148)( 80,147)( 81,137)
( 82,138)( 83,144)( 84,143)( 85,157)( 86,158)( 87,152)( 88,151)( 89,133)
( 90,134)( 91,135)( 92,136)( 93,149)( 94,150)( 95,159)( 96,160)( 97,141)
( 98,142)(101,121)(102,122)(103,123)(104,124)(105,113)(106,114)(107,108)
(109,129)(110,130)(111,116)(112,115)(117,125)(118,126)(119,120)(163,196)
(164,195)(165,218)(166,217)(167,220)(168,219)(169,210)(170,209)(171,203)
(172,204)(173,226)(174,225)(175,211)(176,212)(177,202)(178,201)(179,207)
(180,208)(181,222)(182,221)(183,215)(184,216)(185,198)(186,197)(187,200)
(188,199)(189,214)(190,213)(191,224)(192,223)(193,206)(194,205)(227,292)
(228,291)(229,314)(230,313)(231,316)(232,315)(233,306)(234,305)(235,299)
(236,300)(237,322)(238,321)(239,307)(240,308)(241,298)(242,297)(243,303)
(244,304)(245,318)(246,317)(247,311)(248,312)(249,294)(250,293)(251,296)
(252,295)(253,310)(254,309)(255,320)(256,319)(257,302)(258,301)(259,260)
(261,282)(262,281)(263,284)(264,283)(265,274)(266,273)(269,290)(270,289)
(271,275)(272,276)(277,286)(278,285)(287,288);
s3 := Sym(322)!(  3,132)(  4,131)(  5,149)(  6,150)(  7,152)(  8,151)(  9,137)
( 10,138)( 11,161)( 12,162)( 13,144)( 14,143)( 15,142)( 16,141)( 17,155)
( 18,156)( 19,147)( 20,148)( 21,133)( 22,134)( 23,136)( 24,135)( 25,154)
( 26,153)( 27,145)( 28,146)( 29,159)( 30,160)( 31,157)( 32,158)( 33,139)
( 34,140)( 35,100)( 36, 99)( 37,117)( 38,118)( 39,120)( 40,119)( 41,105)
( 42,106)( 43,129)( 44,130)( 45,112)( 46,111)( 47,110)( 48,109)( 49,123)
( 50,124)( 51,115)( 52,116)( 53,101)( 54,102)( 55,104)( 56,103)( 57,122)
( 58,121)( 59,113)( 60,114)( 61,127)( 62,128)( 63,125)( 64,126)( 65,107)
( 66,108)( 67, 68)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 75, 97)( 76, 98)
( 77, 80)( 78, 79)( 81, 91)( 82, 92)( 89, 90)( 93, 95)( 94, 96)(163,291)
(164,292)(165,310)(166,309)(167,311)(168,312)(169,298)(170,297)(171,322)
(172,321)(173,303)(174,304)(175,301)(176,302)(177,316)(178,315)(179,308)
(180,307)(181,294)(182,293)(183,295)(184,296)(185,313)(186,314)(187,306)
(188,305)(189,320)(190,319)(191,318)(192,317)(193,300)(194,299)(195,259)
(196,260)(197,278)(198,277)(199,279)(200,280)(201,266)(202,265)(203,290)
(204,289)(205,271)(206,272)(207,269)(208,270)(209,284)(210,283)(211,276)
(212,275)(213,262)(214,261)(215,263)(216,264)(217,281)(218,282)(219,274)
(220,273)(221,288)(222,287)(223,286)(224,285)(225,268)(226,267)(229,246)
(230,245)(231,247)(232,248)(233,234)(235,258)(236,257)(237,239)(238,240)
(241,252)(242,251)(243,244)(253,256)(254,255);
poly := sub<Sym(322)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;

to this polytope