Polytope of Type {30,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6}*1800c
if this polytope has a name.
Group : SmallGroup(1800,586)
Rank : 3
Schlafli Type : {30,6}
Number of vertices, edges, etc : 150, 450, 30
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 : {30,3}*900
   3-fold quotients : {10,6}*600b
   6-fold quotients : {10,3}*300
   25-fold quotients : {6,6}*72b
   50-fold quotients : {6,3}*36
   75-fold quotients : {2,6}*24
   150-fold quotients : {2,3}*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=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 5.
      14 facets:
         10 of {6}*12
         4 of {30}*60
      30 vertex figures:
         30 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 5.
      6 facets:
         6 of {30}*60
      30 vertex figures:
         30 of {6}*12

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

Twisty Puzzle