Polytope of Type {12,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30313)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 48, 432, 72
Order of s0s1s2 : 36
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {12,18}*864b
   3-fold quotients : {4,18}*576b, {12,6}*576f
   4-fold quotients : {12,18}*432b, {12,9}*432
   6-fold quotients : {4,18}*288, {12,6}*288b
   8-fold quotients : {6,18}*216b
   9-fold quotients : {4,6}*192b
   12-fold quotients : {4,18}*144a, {4,9}*144, {4,18}*144b, {4,18}*144c, {12,6}*144c, {12,3}*144
   16-fold quotients : {6,9}*108
   18-fold quotients : {4,6}*96
   24-fold quotients : {4,9}*72, {2,18}*72, {6,6}*72b
   36-fold quotients : {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {2,9}*36, {6,3}*36
   72-fold quotients : {4,3}*24, {2,6}*24
   108-fold quotients : {4,2}*16
   144-fold quotients : {2,3}*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=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      36 vertex figures:
         24 of {9}*18
         12 of {18}*36
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      24 vertex figures:
         24 of {18}*36
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      48 facets:
         24 of {6}*12
         24 of {12}*24
      24 vertex figures:
         24 of {18}*36
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      24 vertex figures:
         24 of {18}*36
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 4.
      30 facets:
         6 of {12}*24
         24 of {6}*12
      12 vertex figures:
         12 of {18}*36
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      18 facets:
         18 of {12}*24
      12 vertex figures:
         12 of {18}*36
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         12 of {6}*12
         12 of {12}*24
      18 vertex figures:
         12 of {9}*18
         6 of {18}*36

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

Twisty Puzzle