Polytope of Type {18,12}

Play with this polytope as a twisty puzzle

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

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

Twisty Puzzle