Part of the Atlas of Small Regular Polytopes

Polytope of Type {36,4}

Atlas Canonical Name {36,4}*1152d

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1152,154208)
Rank
3
Schläfli Type
{36,4}
Vertices, edges, …
144, 288, 16
Order of s0s1s2
36
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Self-Petrie

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

12-fold

16-fold

24-fold

32-fold

36-fold

48-fold

72-fold

96-fold

144-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 2

8 facets

96 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s1*s2)^2> of order 2

8 facets

72 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2> of order 2

8 facets

72 vertex figures

P/N, where N=<(s0*s1)^15*(s0*s2*s1)^3> of order 2

8 facets

72 vertex figures

P/N, where N=<(s0*s1)^14*s0*s2*(s1*s0)^2*s2*s1*s0*s1> of order 2

8 facets

72 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*s2*s1*s0*s1*s2> of order 4

4 facets

36 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^16*s0*s2*s1*s0*s1*s2> of order 4

4 facets

36 vertex figures

P/N, where N=<(s1*s2)^2, s0*s1*s2*s1*s0*s2> of order 4

4 facets

60 vertex figures

Representations

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