Polytope of Type {4,36}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,36}*1152d
if this polytope has a name.
Group : SmallGroup(1152,154208)
Rank : 3
Schlafli Type : {4,36}
Number of vertices, edges, etc : 16, 288, 144
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 4
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 : {4,36}*576b, {4,18}*576b, {4,36}*576c
   3-fold quotients : {4,12}*384d
   4-fold quotients : {4,36}*288a, {4,36}*288b, {4,36}*288c, {4,18}*288
   6-fold quotients : {4,12}*192b, {4,6}*192b, {4,12}*192c
   8-fold quotients : {2,36}*144, {4,18}*144a, {4,9}*144, {4,18}*144b, {4,18}*144c
   12-fold quotients : {4,12}*96a, {4,12}*96b, {4,12}*96c, {4,6}*96
   16-fold quotients : {4,9}*72, {2,18}*72
   24-fold quotients : {2,12}*48, {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
   32-fold quotients : {2,9}*36
   36-fold quotients : {4,4}*32
   48-fold quotients : {4,3}*24, {2,6}*24
   72-fold quotients : {2,4}*16, {4,2}*16
   96-fold quotients : {2,3}*12
   144-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 2.
      96 facets:
         48 of {2}*4
         48 of {4}*8
      8 vertex figures:
         8 of {36}*72
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      72 facets:
         72 of {4}*8
      8 vertex figures:
         8 of {36}*72
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      72 facets:
         72 of {4}*8
      8 vertex figures:
         8 of {36}*72
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*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> of order 2.
      72 facets:
         72 of {4}*8
      8 vertex figures:
         8 of {36}*72
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*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> of order 2.
      72 facets:
         72 of {4}*8
      8 vertex figures:
         8 of {36}*72
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*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.
      36 facets:
         36 of {4}*8
      4 vertex figures:
         4 of {36}*72
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*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> of order 4.
      36 facets:
         36 of {4}*8
      4 vertex figures:
         4 of {36}*72
   P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      60 facets:
         12 of {4}*8
         48 of {2}*4
      4 vertex figures:
         4 of {36}*72

Permutation Representation (GAP) :

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

Polytope of Type {4,36}

Twisty Puzzle