Polytope of Type {54,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,12}*1296a
Also Known As : {54,12|2}. if this polytope has another name.
Group : SmallGroup(1296,825)
Rank : 3
Schlafli Type : {54,12}
Number of vertices, edges, etc : 54, 324, 12
Order of s0s1s2 : 108
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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 : {54,6}*648a
   3-fold quotients : {54,4}*432a, {18,12}*432a
   6-fold quotients : {54,2}*216, {18,6}*216a
   9-fold quotients : {18,4}*144a, {6,12}*144a
   12-fold quotients : {27,2}*108
   18-fold quotients : {18,2}*72, {6,6}*72a
   27-fold quotients : {2,12}*48, {6,4}*48a
   36-fold quotients : {9,2}*36
   54-fold quotients : {2,6}*24, {6,2}*24
   81-fold quotients : {2,4}*16
   108-fold quotients : {2,3}*12, {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):
   None.

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4,  9)(  5,  8)(  6,  7)( 11, 12)( 13, 18)( 14, 17)( 15, 16)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 63)( 29, 62)( 30, 61)( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 72)( 38, 71)( 39, 70)( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)( 46, 81)( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)(103,108)(104,107)(105,106)(109,144)(110,143)(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,153)(119,152)(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(164,165)(166,171)(167,170)(168,169)(173,174)(175,180)(176,179)(177,178)(182,183)(184,189)(185,188)(186,187)(190,225)(191,224)(192,223)(193,222)(194,221)(195,220)(196,219)(197,218)(198,217)(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)(208,243)(209,242)(210,241)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(245,246)(247,252)(248,251)(249,250)(254,255)(256,261)(257,260)(258,259)(263,264)(265,270)(266,269)(267,268)(271,306)(272,305)(273,304)(274,303)(275,302)(276,301)(277,300)(278,299)(279,298)(280,315)(281,314)(282,313)(283,312)(284,311)(285,310)(286,309)(287,308)(288,307)(289,324)(290,323)(291,322)(292,321)(293,320)(294,319)(295,318)(296,317)(297,316);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 36)(  5, 35)(  6, 34)(  7, 33)(  8, 32)(  9, 31)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)( 16, 51)( 17, 50)( 18, 49)( 19, 37)( 20, 39)( 21, 38)( 22, 45)( 23, 44)( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)( 90,112)( 91,127)( 92,129)( 93,128)( 94,135)( 95,134)( 96,133)( 97,132)( 98,131)( 99,130)(100,118)(101,120)(102,119)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(136,144)(137,143)(138,142)(139,141)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(163,271)(164,273)(165,272)(166,279)(167,278)(168,277)(169,276)(170,275)(171,274)(172,289)(173,291)(174,290)(175,297)(176,296)(177,295)(178,294)(179,293)(180,292)(181,280)(182,282)(183,281)(184,288)(185,287)(186,286)(187,285)(188,284)(189,283)(190,244)(191,246)(192,245)(193,252)(194,251)(195,250)(196,249)(197,248)(198,247)(199,262)(200,264)(201,263)(202,270)(203,269)(204,268)(205,267)(206,266)(207,265)(208,253)(209,255)(210,254)(211,261)(212,260)(213,259)(214,258)(215,257)(216,256)(217,306)(218,305)(219,304)(220,303)(221,302)(222,301)(223,300)(224,299)(225,298)(226,324)(227,323)(228,322)(229,321)(230,320)(231,319)(232,318)(233,317)(234,316)(235,315)(236,314)(237,313)(238,312)(239,311)(240,310)(241,309)(242,308)(243,307);;
s2 := (  1,172)(  2,173)(  3,174)(  4,175)(  5,176)(  6,177)(  7,178)(  8,179)(  9,180)( 10,163)( 11,164)( 12,165)( 13,166)( 14,167)( 15,168)( 16,169)( 17,170)( 18,171)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)( 25,187)( 26,188)( 27,189)( 28,199)( 29,200)( 30,201)( 31,202)( 32,203)( 33,204)( 34,205)( 35,206)( 36,207)( 37,190)( 38,191)( 39,192)( 40,193)( 41,194)( 42,195)( 43,196)( 44,197)( 45,198)( 46,208)( 47,209)( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,226)( 56,227)( 57,228)( 58,229)( 59,230)( 60,231)( 61,232)( 62,233)( 63,234)( 64,217)( 65,218)( 66,219)( 67,220)( 68,221)( 69,222)( 70,223)( 71,224)( 72,225)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)( 81,243)( 82,253)( 83,254)( 84,255)( 85,256)( 86,257)( 87,258)( 88,259)( 89,260)( 90,261)( 91,244)( 92,245)( 93,246)( 94,247)( 95,248)( 96,249)( 97,250)( 98,251)( 99,252)(100,262)(101,263)(102,264)(103,265)(104,266)(105,267)(106,268)(107,269)(108,270)(109,280)(110,281)(111,282)(112,283)(113,284)(114,285)(115,286)(116,287)(117,288)(118,271)(119,272)(120,273)(121,274)(122,275)(123,276)(124,277)(125,278)(126,279)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,307)(137,308)(138,309)(139,310)(140,311)(141,312)(142,313)(143,314)(144,315)(145,298)(146,299)(147,300)(148,301)(149,302)(150,303)(151,304)(152,305)(153,306)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324);;
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*s2*s1*s0*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*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*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*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(324)!(  2,  3)(  4,  9)(  5,  8)(  6,  7)( 11, 12)( 13, 18)( 14, 17)( 15, 16)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 63)( 29, 62)( 30, 61)( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 72)( 38, 71)( 39, 70)( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)( 46, 81)( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)(103,108)(104,107)(105,106)(109,144)(110,143)(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,153)(119,152)(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(164,165)(166,171)(167,170)(168,169)(173,174)(175,180)(176,179)(177,178)(182,183)(184,189)(185,188)(186,187)(190,225)(191,224)(192,223)(193,222)(194,221)(195,220)(196,219)(197,218)(198,217)(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)(208,243)(209,242)(210,241)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(245,246)(247,252)(248,251)(249,250)(254,255)(256,261)(257,260)(258,259)(263,264)(265,270)(266,269)(267,268)(271,306)(272,305)(273,304)(274,303)(275,302)(276,301)(277,300)(278,299)(279,298)(280,315)(281,314)(282,313)(283,312)(284,311)(285,310)(286,309)(287,308)(288,307)(289,324)(290,323)(291,322)(292,321)(293,320)(294,319)(295,318)(296,317)(297,316);
s1 := Sym(324)!(  1, 28)(  2, 30)(  3, 29)(  4, 36)(  5, 35)(  6, 34)(  7, 33)(  8, 32)(  9, 31)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)( 16, 51)( 17, 50)( 18, 49)( 19, 37)( 20, 39)( 21, 38)( 22, 45)( 23, 44)( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)( 90,112)( 91,127)( 92,129)( 93,128)( 94,135)( 95,134)( 96,133)( 97,132)( 98,131)( 99,130)(100,118)(101,120)(102,119)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(136,144)(137,143)(138,142)(139,141)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(163,271)(164,273)(165,272)(166,279)(167,278)(168,277)(169,276)(170,275)(171,274)(172,289)(173,291)(174,290)(175,297)(176,296)(177,295)(178,294)(179,293)(180,292)(181,280)(182,282)(183,281)(184,288)(185,287)(186,286)(187,285)(188,284)(189,283)(190,244)(191,246)(192,245)(193,252)(194,251)(195,250)(196,249)(197,248)(198,247)(199,262)(200,264)(201,263)(202,270)(203,269)(204,268)(205,267)(206,266)(207,265)(208,253)(209,255)(210,254)(211,261)(212,260)(213,259)(214,258)(215,257)(216,256)(217,306)(218,305)(219,304)(220,303)(221,302)(222,301)(223,300)(224,299)(225,298)(226,324)(227,323)(228,322)(229,321)(230,320)(231,319)(232,318)(233,317)(234,316)(235,315)(236,314)(237,313)(238,312)(239,311)(240,310)(241,309)(242,308)(243,307);
s2 := Sym(324)!(  1,172)(  2,173)(  3,174)(  4,175)(  5,176)(  6,177)(  7,178)(  8,179)(  9,180)( 10,163)( 11,164)( 12,165)( 13,166)( 14,167)( 15,168)( 16,169)( 17,170)( 18,171)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)( 25,187)( 26,188)( 27,189)( 28,199)( 29,200)( 30,201)( 31,202)( 32,203)( 33,204)( 34,205)( 35,206)( 36,207)( 37,190)( 38,191)( 39,192)( 40,193)( 41,194)( 42,195)( 43,196)( 44,197)( 45,198)( 46,208)( 47,209)( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,226)( 56,227)( 57,228)( 58,229)( 59,230)( 60,231)( 61,232)( 62,233)( 63,234)( 64,217)( 65,218)( 66,219)( 67,220)( 68,221)( 69,222)( 70,223)( 71,224)( 72,225)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)( 81,243)( 82,253)( 83,254)( 84,255)( 85,256)( 86,257)( 87,258)( 88,259)( 89,260)( 90,261)( 91,244)( 92,245)( 93,246)( 94,247)( 95,248)( 96,249)( 97,250)( 98,251)( 99,252)(100,262)(101,263)(102,264)(103,265)(104,266)(105,267)(106,268)(107,269)(108,270)(109,280)(110,281)(111,282)(112,283)(113,284)(114,285)(115,286)(116,287)(117,288)(118,271)(119,272)(120,273)(121,274)(122,275)(123,276)(124,277)(125,278)(126,279)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,307)(137,308)(138,309)(139,310)(140,311)(141,312)(142,313)(143,314)(144,315)(145,298)(146,299)(147,300)(148,301)(149,302)(150,303)(151,304)(152,305)(153,306)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324);
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*s2*s1*s0*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*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*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*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