Polytope of Type {6,12}

Play with this polytope as a twisty puzzle

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

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