Polytope of Type {6,30}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*1800d
if this polytope has a name.
Group : SmallGroup(1800,586)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 30, 450, 150
Order of s0s1s2 : 6
Order of s0s1s2s1 : 30
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,30}*900
   3-fold quotients : {6,10}*600b
   6-fold quotients : {3,10}*300
   25-fold quotients : {6,6}*72c
   50-fold quotients : {3,6}*36
   75-fold quotients : {6,2}*24
   150-fold quotients : {3,2}*12
   225-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 5.
      30 facets:
         30 of {6}*12
      14 vertex figures:
         4 of {30}*60
         10 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 5.
      30 facets:
         30 of {6}*12
      6 vertex figures:
         6 of {30}*60

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

Twisty Puzzle