Polytope of Type {126,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {126,4}*1008c
if this polytope has a name.
Group : SmallGroup(1008,503)
Rank : 3
Schlafli Type : {126,4}
Number of vertices, edges, etc : 126, 252, 4
Order of s₀s₁s₂ : 63
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {63,4}*504
   3-fold quotients : {42,4}*336c
   6-fold quotients : {21,4}*168
   7-fold quotients : {18,4}*144c
   14-fold quotients : {9,4}*72
   21-fold quotients : {6,4}*48b
   42-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) :

s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 73)( 14, 74)( 15, 76)( 16, 75)( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 77)( 22, 78)( 23, 80)( 24, 79)( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 69)( 30, 70)( 31, 72)( 32, 71)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 41, 57)( 42, 58)( 43, 60)( 44, 59)( 45, 53)( 46, 54)( 47, 56)( 48, 55)( 85,177)( 86,178)( 87,180)( 88,179)( 89,173)( 90,174)( 91,176)( 92,175)( 93,169)( 94,170)( 95,172)( 96,171)( 97,249)( 98,250)( 99,252)(100,251)(101,245)(102,246)(103,248)(104,247)(105,241)(106,242)(107,244)(108,243)(109,237)(110,238)(111,240)(112,239)(113,233)(114,234)(115,236)(116,235)(117,229)(118,230)(119,232)(120,231)(121,225)(122,226)(123,228)(124,227)(125,221)(126,222)(127,224)(128,223)(129,217)(130,218)(131,220)(132,219)(133,213)(134,214)(135,216)(136,215)(137,209)(138,210)(139,212)(140,211)(141,205)(142,206)(143,208)(144,207)(145,201)(146,202)(147,204)(148,203)(149,197)(150,198)(151,200)(152,199)(153,193)(154,194)(155,196)(156,195)(157,189)(158,190)(159,192)(160,191)(161,185)(162,186)(163,188)(164,187)(165,181)(166,182)(167,184)(168,183)(255,256)(257,261)(258,262)(259,264)(260,263)(265,325)(266,326)(267,328)(268,327)(269,333)(270,334)(271,336)(272,335)(273,329)(274,330)(275,332)(276,331)(277,313)(278,314)(279,316)(280,315)(281,321)(282,322)(283,324)(284,323)(285,317)(286,318)(287,320)(288,319)(289,301)(290,302)(291,304)(292,303)(293,309)(294,310)(295,312)(296,311)(297,305)(298,306)(299,308)(300,307)(337,429)(338,430)(339,432)(340,431)(341,425)(342,426)(343,428)(344,427)(345,421)(346,422)(347,424)(348,423)(349,501)(350,502)(351,504)(352,503)(353,497)(354,498)(355,500)(356,499)(357,493)(358,494)(359,496)(360,495)(361,489)(362,490)(363,492)(364,491)(365,485)(366,486)(367,488)(368,487)(369,481)(370,482)(371,484)(372,483)(373,477)(374,478)(375,480)(376,479)(377,473)(378,474)(379,476)(380,475)(381,469)(382,470)(383,472)(384,471)(385,465)(386,466)(387,468)(388,467)(389,461)(390,462)(391,464)(392,463)(393,457)(394,458)(395,460)(396,459)(397,453)(398,454)(399,456)(400,455)(401,449)(402,450)(403,452)(404,451)(405,445)(406,446)(407,448)(408,447)(409,441)(410,442)(411,444)(412,443)(413,437)(414,438)(415,440)(416,439)(417,433)(418,434)(419,436)(420,435);;
s1 := (  1,349)(  2,352)(  3,351)(  4,350)(  5,357)(  6,360)(  7,359)(  8,358)(  9,353)( 10,356)( 11,355)( 12,354)( 13,337)( 14,340)( 15,339)( 16,338)( 17,345)( 18,348)( 19,347)( 20,346)( 21,341)( 22,344)( 23,343)( 24,342)( 25,409)( 26,412)( 27,411)( 28,410)( 29,417)( 30,420)( 31,419)( 32,418)( 33,413)( 34,416)( 35,415)( 36,414)( 37,397)( 38,400)( 39,399)( 40,398)( 41,405)( 42,408)( 43,407)( 44,406)( 45,401)( 46,404)( 47,403)( 48,402)( 49,385)( 50,388)( 51,387)( 52,386)( 53,393)( 54,396)( 55,395)( 56,394)( 57,389)( 58,392)( 59,391)( 60,390)( 61,373)( 62,376)( 63,375)( 64,374)( 65,381)( 66,384)( 67,383)( 68,382)( 69,377)( 70,380)( 71,379)( 72,378)( 73,361)( 74,364)( 75,363)( 76,362)( 77,369)( 78,372)( 79,371)( 80,370)( 81,365)( 82,368)( 83,367)( 84,366)( 85,265)( 86,268)( 87,267)( 88,266)( 89,273)( 90,276)( 91,275)( 92,274)( 93,269)( 94,272)( 95,271)( 96,270)( 97,253)( 98,256)( 99,255)(100,254)(101,261)(102,264)(103,263)(104,262)(105,257)(106,260)(107,259)(108,258)(109,325)(110,328)(111,327)(112,326)(113,333)(114,336)(115,335)(116,334)(117,329)(118,332)(119,331)(120,330)(121,313)(122,316)(123,315)(124,314)(125,321)(126,324)(127,323)(128,322)(129,317)(130,320)(131,319)(132,318)(133,301)(134,304)(135,303)(136,302)(137,309)(138,312)(139,311)(140,310)(141,305)(142,308)(143,307)(144,306)(145,289)(146,292)(147,291)(148,290)(149,297)(150,300)(151,299)(152,298)(153,293)(154,296)(155,295)(156,294)(157,277)(158,280)(159,279)(160,278)(161,285)(162,288)(163,287)(164,286)(165,281)(166,284)(167,283)(168,282)(169,441)(170,444)(171,443)(172,442)(173,437)(174,440)(175,439)(176,438)(177,433)(178,436)(179,435)(180,434)(181,429)(182,432)(183,431)(184,430)(185,425)(186,428)(187,427)(188,426)(189,421)(190,424)(191,423)(192,422)(193,501)(194,504)(195,503)(196,502)(197,497)(198,500)(199,499)(200,498)(201,493)(202,496)(203,495)(204,494)(205,489)(206,492)(207,491)(208,490)(209,485)(210,488)(211,487)(212,486)(213,481)(214,484)(215,483)(216,482)(217,477)(218,480)(219,479)(220,478)(221,473)(222,476)(223,475)(224,474)(225,469)(226,472)(227,471)(228,470)(229,465)(230,468)(231,467)(232,466)(233,461)(234,464)(235,463)(236,462)(237,457)(238,460)(239,459)(240,458)(241,453)(242,456)(243,455)(244,454)(245,449)(246,452)(247,451)(248,450)(249,445)(250,448)(251,447)(252,446);;
s2 := (  1,254)(  2,253)(  3,256)(  4,255)(  5,258)(  6,257)(  7,260)(  8,259)(  9,262)( 10,261)( 11,264)( 12,263)( 13,266)( 14,265)( 15,268)( 16,267)( 17,270)( 18,269)( 19,272)( 20,271)( 21,274)( 22,273)( 23,276)( 24,275)( 25,278)( 26,277)( 27,280)( 28,279)( 29,282)( 30,281)( 31,284)( 32,283)( 33,286)( 34,285)( 35,288)( 36,287)( 37,290)( 38,289)( 39,292)( 40,291)( 41,294)( 42,293)( 43,296)( 44,295)( 45,298)( 46,297)( 47,300)( 48,299)( 49,302)( 50,301)( 51,304)( 52,303)( 53,306)( 54,305)( 55,308)( 56,307)( 57,310)( 58,309)( 59,312)( 60,311)( 61,314)( 62,313)( 63,316)( 64,315)( 65,318)( 66,317)( 67,320)( 68,319)( 69,322)( 70,321)( 71,324)( 72,323)( 73,326)( 74,325)( 75,328)( 76,327)( 77,330)( 78,329)( 79,332)( 80,331)( 81,334)( 82,333)( 83,336)( 84,335)( 85,338)( 86,337)( 87,340)( 88,339)( 89,342)( 90,341)( 91,344)( 92,343)( 93,346)( 94,345)( 95,348)( 96,347)( 97,350)( 98,349)( 99,352)(100,351)(101,354)(102,353)(103,356)(104,355)(105,358)(106,357)(107,360)(108,359)(109,362)(110,361)(111,364)(112,363)(113,366)(114,365)(115,368)(116,367)(117,370)(118,369)(119,372)(120,371)(121,374)(122,373)(123,376)(124,375)(125,378)(126,377)(127,380)(128,379)(129,382)(130,381)(131,384)(132,383)(133,386)(134,385)(135,388)(136,387)(137,390)(138,389)(139,392)(140,391)(141,394)(142,393)(143,396)(144,395)(145,398)(146,397)(147,400)(148,399)(149,402)(150,401)(151,404)(152,403)(153,406)(154,405)(155,408)(156,407)(157,410)(158,409)(159,412)(160,411)(161,414)(162,413)(163,416)(164,415)(165,418)(166,417)(167,420)(168,419)(169,422)(170,421)(171,424)(172,423)(173,426)(174,425)(175,428)(176,427)(177,430)(178,429)(179,432)(180,431)(181,434)(182,433)(183,436)(184,435)(185,438)(186,437)(187,440)(188,439)(189,442)(190,441)(191,444)(192,443)(193,446)(194,445)(195,448)(196,447)(197,450)(198,449)(199,452)(200,451)(201,454)(202,453)(203,456)(204,455)(205,458)(206,457)(207,460)(208,459)(209,462)(210,461)(211,464)(212,463)(213,466)(214,465)(215,468)(216,467)(217,470)(218,469)(219,472)(220,471)(221,474)(222,473)(223,476)(224,475)(225,478)(226,477)(227,480)(228,479)(229,482)(230,481)(231,484)(232,483)(233,486)(234,485)(235,488)(236,487)(237,490)(238,489)(239,492)(240,491)(241,494)(242,493)(243,496)(244,495)(245,498)(246,497)(247,500)(248,499)(249,502)(250,501)(251,504)(252,503);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s1*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*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(504)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 73)( 14, 74)( 15, 76)( 16, 75)( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 77)( 22, 78)( 23, 80)( 24, 79)( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 69)( 30, 70)( 31, 72)( 32, 71)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 41, 57)( 42, 58)( 43, 60)( 44, 59)( 45, 53)( 46, 54)( 47, 56)( 48, 55)( 85,177)( 86,178)( 87,180)( 88,179)( 89,173)( 90,174)( 91,176)( 92,175)( 93,169)( 94,170)( 95,172)( 96,171)( 97,249)( 98,250)( 99,252)(100,251)(101,245)(102,246)(103,248)(104,247)(105,241)(106,242)(107,244)(108,243)(109,237)(110,238)(111,240)(112,239)(113,233)(114,234)(115,236)(116,235)(117,229)(118,230)(119,232)(120,231)(121,225)(122,226)(123,228)(124,227)(125,221)(126,222)(127,224)(128,223)(129,217)(130,218)(131,220)(132,219)(133,213)(134,214)(135,216)(136,215)(137,209)(138,210)(139,212)(140,211)(141,205)(142,206)(143,208)(144,207)(145,201)(146,202)(147,204)(148,203)(149,197)(150,198)(151,200)(152,199)(153,193)(154,194)(155,196)(156,195)(157,189)(158,190)(159,192)(160,191)(161,185)(162,186)(163,188)(164,187)(165,181)(166,182)(167,184)(168,183)(255,256)(257,261)(258,262)(259,264)(260,263)(265,325)(266,326)(267,328)(268,327)(269,333)(270,334)(271,336)(272,335)(273,329)(274,330)(275,332)(276,331)(277,313)(278,314)(279,316)(280,315)(281,321)(282,322)(283,324)(284,323)(285,317)(286,318)(287,320)(288,319)(289,301)(290,302)(291,304)(292,303)(293,309)(294,310)(295,312)(296,311)(297,305)(298,306)(299,308)(300,307)(337,429)(338,430)(339,432)(340,431)(341,425)(342,426)(343,428)(344,427)(345,421)(346,422)(347,424)(348,423)(349,501)(350,502)(351,504)(352,503)(353,497)(354,498)(355,500)(356,499)(357,493)(358,494)(359,496)(360,495)(361,489)(362,490)(363,492)(364,491)(365,485)(366,486)(367,488)(368,487)(369,481)(370,482)(371,484)(372,483)(373,477)(374,478)(375,480)(376,479)(377,473)(378,474)(379,476)(380,475)(381,469)(382,470)(383,472)(384,471)(385,465)(386,466)(387,468)(388,467)(389,461)(390,462)(391,464)(392,463)(393,457)(394,458)(395,460)(396,459)(397,453)(398,454)(399,456)(400,455)(401,449)(402,450)(403,452)(404,451)(405,445)(406,446)(407,448)(408,447)(409,441)(410,442)(411,444)(412,443)(413,437)(414,438)(415,440)(416,439)(417,433)(418,434)(419,436)(420,435);
s1 := Sym(504)!(  1,349)(  2,352)(  3,351)(  4,350)(  5,357)(  6,360)(  7,359)(  8,358)(  9,353)( 10,356)( 11,355)( 12,354)( 13,337)( 14,340)( 15,339)( 16,338)( 17,345)( 18,348)( 19,347)( 20,346)( 21,341)( 22,344)( 23,343)( 24,342)( 25,409)( 26,412)( 27,411)( 28,410)( 29,417)( 30,420)( 31,419)( 32,418)( 33,413)( 34,416)( 35,415)( 36,414)( 37,397)( 38,400)( 39,399)( 40,398)( 41,405)( 42,408)( 43,407)( 44,406)( 45,401)( 46,404)( 47,403)( 48,402)( 49,385)( 50,388)( 51,387)( 52,386)( 53,393)( 54,396)( 55,395)( 56,394)( 57,389)( 58,392)( 59,391)( 60,390)( 61,373)( 62,376)( 63,375)( 64,374)( 65,381)( 66,384)( 67,383)( 68,382)( 69,377)( 70,380)( 71,379)( 72,378)( 73,361)( 74,364)( 75,363)( 76,362)( 77,369)( 78,372)( 79,371)( 80,370)( 81,365)( 82,368)( 83,367)( 84,366)( 85,265)( 86,268)( 87,267)( 88,266)( 89,273)( 90,276)( 91,275)( 92,274)( 93,269)( 94,272)( 95,271)( 96,270)( 97,253)( 98,256)( 99,255)(100,254)(101,261)(102,264)(103,263)(104,262)(105,257)(106,260)(107,259)(108,258)(109,325)(110,328)(111,327)(112,326)(113,333)(114,336)(115,335)(116,334)(117,329)(118,332)(119,331)(120,330)(121,313)(122,316)(123,315)(124,314)(125,321)(126,324)(127,323)(128,322)(129,317)(130,320)(131,319)(132,318)(133,301)(134,304)(135,303)(136,302)(137,309)(138,312)(139,311)(140,310)(141,305)(142,308)(143,307)(144,306)(145,289)(146,292)(147,291)(148,290)(149,297)(150,300)(151,299)(152,298)(153,293)(154,296)(155,295)(156,294)(157,277)(158,280)(159,279)(160,278)(161,285)(162,288)(163,287)(164,286)(165,281)(166,284)(167,283)(168,282)(169,441)(170,444)(171,443)(172,442)(173,437)(174,440)(175,439)(176,438)(177,433)(178,436)(179,435)(180,434)(181,429)(182,432)(183,431)(184,430)(185,425)(186,428)(187,427)(188,426)(189,421)(190,424)(191,423)(192,422)(193,501)(194,504)(195,503)(196,502)(197,497)(198,500)(199,499)(200,498)(201,493)(202,496)(203,495)(204,494)(205,489)(206,492)(207,491)(208,490)(209,485)(210,488)(211,487)(212,486)(213,481)(214,484)(215,483)(216,482)(217,477)(218,480)(219,479)(220,478)(221,473)(222,476)(223,475)(224,474)(225,469)(226,472)(227,471)(228,470)(229,465)(230,468)(231,467)(232,466)(233,461)(234,464)(235,463)(236,462)(237,457)(238,460)(239,459)(240,458)(241,453)(242,456)(243,455)(244,454)(245,449)(246,452)(247,451)(248,450)(249,445)(250,448)(251,447)(252,446);
s2 := Sym(504)!(  1,254)(  2,253)(  3,256)(  4,255)(  5,258)(  6,257)(  7,260)(  8,259)(  9,262)( 10,261)( 11,264)( 12,263)( 13,266)( 14,265)( 15,268)( 16,267)( 17,270)( 18,269)( 19,272)( 20,271)( 21,274)( 22,273)( 23,276)( 24,275)( 25,278)( 26,277)( 27,280)( 28,279)( 29,282)( 30,281)( 31,284)( 32,283)( 33,286)( 34,285)( 35,288)( 36,287)( 37,290)( 38,289)( 39,292)( 40,291)( 41,294)( 42,293)( 43,296)( 44,295)( 45,298)( 46,297)( 47,300)( 48,299)( 49,302)( 50,301)( 51,304)( 52,303)( 53,306)( 54,305)( 55,308)( 56,307)( 57,310)( 58,309)( 59,312)( 60,311)( 61,314)( 62,313)( 63,316)( 64,315)( 65,318)( 66,317)( 67,320)( 68,319)( 69,322)( 70,321)( 71,324)( 72,323)( 73,326)( 74,325)( 75,328)( 76,327)( 77,330)( 78,329)( 79,332)( 80,331)( 81,334)( 82,333)( 83,336)( 84,335)( 85,338)( 86,337)( 87,340)( 88,339)( 89,342)( 90,341)( 91,344)( 92,343)( 93,346)( 94,345)( 95,348)( 96,347)( 97,350)( 98,349)( 99,352)(100,351)(101,354)(102,353)(103,356)(104,355)(105,358)(106,357)(107,360)(108,359)(109,362)(110,361)(111,364)(112,363)(113,366)(114,365)(115,368)(116,367)(117,370)(118,369)(119,372)(120,371)(121,374)(122,373)(123,376)(124,375)(125,378)(126,377)(127,380)(128,379)(129,382)(130,381)(131,384)(132,383)(133,386)(134,385)(135,388)(136,387)(137,390)(138,389)(139,392)(140,391)(141,394)(142,393)(143,396)(144,395)(145,398)(146,397)(147,400)(148,399)(149,402)(150,401)(151,404)(152,403)(153,406)(154,405)(155,408)(156,407)(157,410)(158,409)(159,412)(160,411)(161,414)(162,413)(163,416)(164,415)(165,418)(166,417)(167,420)(168,419)(169,422)(170,421)(171,424)(172,423)(173,426)(174,425)(175,428)(176,427)(177,430)(178,429)(179,432)(180,431)(181,434)(182,433)(183,436)(184,435)(185,438)(186,437)(187,440)(188,439)(189,442)(190,441)(191,444)(192,443)(193,446)(194,445)(195,448)(196,447)(197,450)(198,449)(199,452)(200,451)(201,454)(202,453)(203,456)(204,455)(205,458)(206,457)(207,460)(208,459)(209,462)(210,461)(211,464)(212,463)(213,466)(214,465)(215,468)(216,467)(217,470)(218,469)(219,472)(220,471)(221,474)(222,473)(223,476)(224,475)(225,478)(226,477)(227,480)(228,479)(229,482)(230,481)(231,484)(232,483)(233,486)(234,485)(235,488)(236,487)(237,490)(238,489)(239,492)(240,491)(241,494)(242,493)(243,496)(244,495)(245,498)(246,497)(247,500)(248,499)(249,502)(250,501)(251,504)(252,503);
poly := sub<Sym(504)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s1*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*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

Polytope of Type {126,4}

Twisty Puzzle