Polytope of Type {72,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30189)
Rank : 3
Schlafli Type : {72,6}
Number of vertices, edges, etc : 144, 432, 12
Order of s0s1s2 : 18
Order of s0s1s2s1 : 8
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 : {36,6}*864
   3-fold quotients : {24,6}*576c
   4-fold quotients : {36,6}*432c
   6-fold quotients : {12,6}*288a
   8-fold quotients : {18,6}*216a
   9-fold quotients : {8,6}*192c
   12-fold quotients : {12,6}*144d
   18-fold quotients : {4,6}*96
   24-fold quotients : {18,2}*72, {6,6}*72a
   36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {9,2}*36
   72-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
   144-fold quotients : {2,3}*12, {3,2}*12
   216-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*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 2.
      6 facets:
         6 of {72}*144
      72 vertex figures:
         72 of {6}*12

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

Twisty Puzzle