Polytope of Type {12,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296c
if this polytope has a name.
Group : SmallGroup(1296,868)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
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 : {6,6}*648d
   3-fold quotients : {12,6}*432a
   4-fold quotients : {3,6}*324
   6-fold quotients : {6,6}*216c
   9-fold quotients : {12,6}*144b
   12-fold quotients : {3,6}*108
   18-fold quotients : {6,6}*72c
   27-fold quotients : {12,2}*48
   36-fold quotients : {3,6}*36
   54-fold quotients : {6,2}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {3,2}*12
   162-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*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      44 vertex figures:
         32 of {6}*12
         12 of {2}*4
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4,  6)(  7,  8)( 10, 20)( 11, 19)( 12, 21)( 13, 22)( 14, 24)( 15, 23)( 16, 27)( 17, 26)( 18, 25)( 28, 55)( 29, 57)( 30, 56)( 31, 60)( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 81)( 44, 80)( 45, 79)( 46, 65)( 47, 64)( 48, 66)( 49, 67)( 50, 69)( 51, 68)( 52, 72)( 53, 71)( 54, 70)( 83, 84)( 85, 87)( 88, 89)( 91,101)( 92,100)( 93,102)( 94,103)( 95,105)( 96,104)( 97,108)( 98,107)( 99,106)(109,136)(110,138)(111,137)(112,141)(113,140)(114,139)(115,143)(116,142)(117,144)(118,155)(119,154)(120,156)(121,157)(122,159)(123,158)(124,162)(125,161)(126,160)(127,146)(128,145)(129,147)(130,148)(131,150)(132,149)(133,153)(134,152)(135,151)(163,244)(164,246)(165,245)(166,249)(167,248)(168,247)(169,251)(170,250)(171,252)(172,263)(173,262)(174,264)(175,265)(176,267)(177,266)(178,270)(179,269)(180,268)(181,254)(182,253)(183,255)(184,256)(185,258)(186,257)(187,261)(188,260)(189,259)(190,298)(191,300)(192,299)(193,303)(194,302)(195,301)(196,305)(197,304)(198,306)(199,317)(200,316)(201,318)(202,319)(203,321)(204,320)(205,324)(206,323)(207,322)(208,308)(209,307)(210,309)(211,310)(212,312)(213,311)(214,315)(215,314)(216,313)(217,271)(218,273)(219,272)(220,276)(221,275)(222,274)(223,278)(224,277)(225,279)(226,290)(227,289)(228,291)(229,292)(230,294)(231,293)(232,297)(233,296)(234,295)(235,281)(236,280)(237,282)(238,283)(239,285)(240,284)(241,288)(242,287)(243,286);;
s1 := (  1,229)(  2,231)(  3,230)(  4,233)(  5,232)(  6,234)(  7,228)(  8,227)(  9,226)( 10,225)( 11,224)( 12,223)( 13,217)( 14,219)( 15,218)( 16,221)( 17,220)( 18,222)( 19,237)( 20,236)( 21,235)( 22,238)( 23,240)( 24,239)( 25,242)( 26,241)( 27,243)( 28,202)( 29,204)( 30,203)( 31,206)( 32,205)( 33,207)( 34,201)( 35,200)( 36,199)( 37,198)( 38,197)( 39,196)( 40,190)( 41,192)( 42,191)( 43,194)( 44,193)( 45,195)( 46,210)( 47,209)( 48,208)( 49,211)( 50,213)( 51,212)( 52,215)( 53,214)( 54,216)( 55,175)( 56,177)( 57,176)( 58,179)( 59,178)( 60,180)( 61,174)( 62,173)( 63,172)( 64,171)( 65,170)( 66,169)( 67,163)( 68,165)( 69,164)( 70,167)( 71,166)( 72,168)( 73,183)( 74,182)( 75,181)( 76,184)( 77,186)( 78,185)( 79,188)( 80,187)( 81,189)( 82,310)( 83,312)( 84,311)( 85,314)( 86,313)( 87,315)( 88,309)( 89,308)( 90,307)( 91,306)( 92,305)( 93,304)( 94,298)( 95,300)( 96,299)( 97,302)( 98,301)( 99,303)(100,318)(101,317)(102,316)(103,319)(104,321)(105,320)(106,323)(107,322)(108,324)(109,283)(110,285)(111,284)(112,287)(113,286)(114,288)(115,282)(116,281)(117,280)(118,279)(119,278)(120,277)(121,271)(122,273)(123,272)(124,275)(125,274)(126,276)(127,291)(128,290)(129,289)(130,292)(131,294)(132,293)(133,296)(134,295)(135,297)(136,256)(137,258)(138,257)(139,260)(140,259)(141,261)(142,255)(143,254)(144,253)(145,252)(146,251)(147,250)(148,244)(149,246)(150,245)(151,248)(152,247)(153,249)(154,264)(155,263)(156,262)(157,265)(158,267)(159,266)(160,269)(161,268)(162,270);;
s2 := (  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)( 23, 27)( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)( 34, 60)( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)( 42, 70)( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)( 50, 81)( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)( 94, 98)( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)(111,138)(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)(119,146)(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)(127,154)(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)(135,158)(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)(185,189)(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)(196,222)(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)(204,232)(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)(212,243)(213,241)(214,240)(215,238)(216,239)(247,251)(248,252)(249,250)(256,260)(257,261)(258,259)(265,269)(266,270)(267,268)(271,298)(272,299)(273,300)(274,305)(275,306)(276,304)(277,303)(278,301)(279,302)(280,307)(281,308)(282,309)(283,314)(284,315)(285,313)(286,312)(287,310)(288,311)(289,316)(290,317)(291,318)(292,323)(293,324)(294,322)(295,321)(296,319)(297,320);;
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*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(324)!(  2,  3)(  4,  6)(  7,  8)( 10, 20)( 11, 19)( 12, 21)( 13, 22)( 14, 24)( 15, 23)( 16, 27)( 17, 26)( 18, 25)( 28, 55)( 29, 57)( 30, 56)( 31, 60)( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 81)( 44, 80)( 45, 79)( 46, 65)( 47, 64)( 48, 66)( 49, 67)( 50, 69)( 51, 68)( 52, 72)( 53, 71)( 54, 70)( 83, 84)( 85, 87)( 88, 89)( 91,101)( 92,100)( 93,102)( 94,103)( 95,105)( 96,104)( 97,108)( 98,107)( 99,106)(109,136)(110,138)(111,137)(112,141)(113,140)(114,139)(115,143)(116,142)(117,144)(118,155)(119,154)(120,156)(121,157)(122,159)(123,158)(124,162)(125,161)(126,160)(127,146)(128,145)(129,147)(130,148)(131,150)(132,149)(133,153)(134,152)(135,151)(163,244)(164,246)(165,245)(166,249)(167,248)(168,247)(169,251)(170,250)(171,252)(172,263)(173,262)(174,264)(175,265)(176,267)(177,266)(178,270)(179,269)(180,268)(181,254)(182,253)(183,255)(184,256)(185,258)(186,257)(187,261)(188,260)(189,259)(190,298)(191,300)(192,299)(193,303)(194,302)(195,301)(196,305)(197,304)(198,306)(199,317)(200,316)(201,318)(202,319)(203,321)(204,320)(205,324)(206,323)(207,322)(208,308)(209,307)(210,309)(211,310)(212,312)(213,311)(214,315)(215,314)(216,313)(217,271)(218,273)(219,272)(220,276)(221,275)(222,274)(223,278)(224,277)(225,279)(226,290)(227,289)(228,291)(229,292)(230,294)(231,293)(232,297)(233,296)(234,295)(235,281)(236,280)(237,282)(238,283)(239,285)(240,284)(241,288)(242,287)(243,286);
s1 := Sym(324)!(  1,229)(  2,231)(  3,230)(  4,233)(  5,232)(  6,234)(  7,228)(  8,227)(  9,226)( 10,225)( 11,224)( 12,223)( 13,217)( 14,219)( 15,218)( 16,221)( 17,220)( 18,222)( 19,237)( 20,236)( 21,235)( 22,238)( 23,240)( 24,239)( 25,242)( 26,241)( 27,243)( 28,202)( 29,204)( 30,203)( 31,206)( 32,205)( 33,207)( 34,201)( 35,200)( 36,199)( 37,198)( 38,197)( 39,196)( 40,190)( 41,192)( 42,191)( 43,194)( 44,193)( 45,195)( 46,210)( 47,209)( 48,208)( 49,211)( 50,213)( 51,212)( 52,215)( 53,214)( 54,216)( 55,175)( 56,177)( 57,176)( 58,179)( 59,178)( 60,180)( 61,174)( 62,173)( 63,172)( 64,171)( 65,170)( 66,169)( 67,163)( 68,165)( 69,164)( 70,167)( 71,166)( 72,168)( 73,183)( 74,182)( 75,181)( 76,184)( 77,186)( 78,185)( 79,188)( 80,187)( 81,189)( 82,310)( 83,312)( 84,311)( 85,314)( 86,313)( 87,315)( 88,309)( 89,308)( 90,307)( 91,306)( 92,305)( 93,304)( 94,298)( 95,300)( 96,299)( 97,302)( 98,301)( 99,303)(100,318)(101,317)(102,316)(103,319)(104,321)(105,320)(106,323)(107,322)(108,324)(109,283)(110,285)(111,284)(112,287)(113,286)(114,288)(115,282)(116,281)(117,280)(118,279)(119,278)(120,277)(121,271)(122,273)(123,272)(124,275)(125,274)(126,276)(127,291)(128,290)(129,289)(130,292)(131,294)(132,293)(133,296)(134,295)(135,297)(136,256)(137,258)(138,257)(139,260)(140,259)(141,261)(142,255)(143,254)(144,253)(145,252)(146,251)(147,250)(148,244)(149,246)(150,245)(151,248)(152,247)(153,249)(154,264)(155,263)(156,262)(157,265)(158,267)(159,266)(160,269)(161,268)(162,270);
s2 := Sym(324)!(  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)( 23, 27)( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)( 34, 60)( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)( 42, 70)( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)( 50, 81)( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)( 94, 98)( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)(111,138)(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)(119,146)(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)(127,154)(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)(135,158)(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)(185,189)(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)(196,222)(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)(204,232)(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)(212,243)(213,241)(214,240)(215,238)(216,239)(247,251)(248,252)(249,250)(256,260)(257,261)(258,259)(265,269)(266,270)(267,268)(271,298)(272,299)(273,300)(274,305)(275,306)(276,304)(277,303)(278,301)(279,302)(280,307)(281,308)(282,309)(283,314)(284,315)(285,313)(286,312)(287,310)(288,311)(289,316)(290,317)(291,318)(292,323)(293,324)(294,322)(295,321)(296,319)(297,320);
poly := sub<Sym(324)|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*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;
References : None.
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