Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,18}

Atlas Canonical Name {12,18}*1296g

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,917)
Rank
3
Schläfli Type
{12,18}
Vertices, edges, …
36, 324, 54
Order of s0s1s2
12
Order of s0s1s2s1
6
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

9-fold

12-fold

18-fold

27-fold

36-fold

54-fold

81-fold

108-fold

162-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*s1)^2*s0*s2*s1*s0*s1*s2> of order 3

18 facets

12 vertex figures

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

30 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle