Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,18}

Atlas Canonical Name {8,18}*1152g

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1152,154366)
Rank
3
Schläfli Type
{8,18}
Vertices, edges, …
32, 288, 72
Order of s0s1s2
72
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

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=<s0*(s2*s1)^2*s0*(s1*s2)^2> of order 2

36 facets

16 vertex figures

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

36 facets

16 vertex figures

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

36 facets

16 vertex figures

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

18 facets

8 vertex figures

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

18 facets

8 vertex figures

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2, (s0*s2*s1)^3*s0*(s2*s1)^5*s2> of order 4

18 facets

8 vertex figures

Representations

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)( 73,111)( 74,112)( 75,109)( 76,110)( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)( 84,118)( 85,123)( 86,124)( 87,121)( 88,122)( 89,127)( 90,128)( 91,125)( 92,126)( 93,131)( 94,132)( 95,129)( 96,130)( 97,135)( 98,136)( 99,133)(100,134)(101,139)(102,140)(103,137)(104,138)(105,143)(106,144)(107,141)(108,142)(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)( 73,253)( 74,254)( 75,256)( 76,255)( 77,261)( 78,262)( 79,264)( 80,263)( 81,257)( 82,258)( 83,260)( 84,259)( 85,285)( 86,286)( 87,288)( 88,287)( 89,281)( 90,282)( 91,284)( 92,283)( 93,277)( 94,278)( 95,280)( 96,279)( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)(104,271)(105,265)(106,266)(107,268)(108,267)(109,217)(110,218)(111,220)(112,219)(113,225)(114,226)(115,228)(116,227)(117,221)(118,222)(119,224)(120,223)(121,249)(122,250)(123,252)(124,251)(125,245)(126,246)(127,248)(128,247)(129,241)(130,242)(131,244)(132,243)(133,237)(134,238)(135,240)(136,239)(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)(144,231);;
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)(145,157)(146,160)(147,159)(148,158)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)(156,162)(169,177)(170,180)(171,179)(172,178)(174,176)(181,193)(182,196)(183,195)(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)(191,199)(192,198)(205,213)(206,216)(207,215)(208,214)(210,212)(217,229)(218,232)(219,231)(220,230)(221,237)(222,240)(223,239)(224,238)(225,233)(226,236)(227,235)(228,234)(241,249)(242,252)(243,251)(244,250)(246,248)(253,265)(254,268)(255,267)(256,266)(257,273)(258,276)(259,275)(260,274)(261,269)(262,272)(263,271)(264,270)(277,285)(278,288)(279,287)(280,286)(282,284);;
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*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*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*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)( 73,111)( 74,112)( 75,109)( 76,110)( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)( 84,118)( 85,123)( 86,124)( 87,121)( 88,122)( 89,127)( 90,128)( 91,125)( 92,126)( 93,131)( 94,132)( 95,129)( 96,130)( 97,135)( 98,136)( 99,133)(100,134)(101,139)(102,140)(103,137)(104,138)(105,143)(106,144)(107,141)(108,142)(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)( 73,253)( 74,254)( 75,256)( 76,255)( 77,261)( 78,262)( 79,264)( 80,263)( 81,257)( 82,258)( 83,260)( 84,259)( 85,285)( 86,286)( 87,288)( 88,287)( 89,281)( 90,282)( 91,284)( 92,283)( 93,277)( 94,278)( 95,280)( 96,279)( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)(104,271)(105,265)(106,266)(107,268)(108,267)(109,217)(110,218)(111,220)(112,219)(113,225)(114,226)(115,228)(116,227)(117,221)(118,222)(119,224)(120,223)(121,249)(122,250)(123,252)(124,251)(125,245)(126,246)(127,248)(128,247)(129,241)(130,242)(131,244)(132,243)(133,237)(134,238)(135,240)(136,239)(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)(144,231);
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)(145,157)(146,160)(147,159)(148,158)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)(156,162)(169,177)(170,180)(171,179)(172,178)(174,176)(181,193)(182,196)(183,195)(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)(191,199)(192,198)(205,213)(206,216)(207,215)(208,214)(210,212)(217,229)(218,232)(219,231)(220,230)(221,237)(222,240)(223,239)(224,238)(225,233)(226,236)(227,235)(228,234)(241,249)(242,252)(243,251)(244,250)(246,248)(253,265)(254,268)(255,267)(256,266)(257,273)(258,276)(259,275)(260,274)(261,269)(262,272)(263,271)(264,270)(277,285)(278,288)(279,287)(280,286)(282,284);
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References

None.

to this polytope.

Twisty Puzzle