Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,12}

Atlas Canonical Name {3,12}*1296a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,1786)
Rank
3
Schläfli Type
{3,12}
Vertices, edges, …
54, 324, 216
Order of s0s1s2
18
Order of s0s1s2s1
12
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

4-fold

9-fold

12-fold

27-fold

36-fold

54-fold

108-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*s0*(s2*s1)^5*s0*s2*s1> of order 2

108 facets

36 vertex figures

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

108 facets

27 vertex figures

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

72 facets

18 vertex figures

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

72 facets

22 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1)^5*s0*s2*s1, s1*s0*s2*s1*s0*(s2*s1)^2*(s0*s2*s1)^2*s2> of order 6

36 facets

16 vertex figures

P/N, where N=<(s1*s2)^4, s0*s1*s0*(s2*s1)^5*s0*s2*s1> of order 6

36 facets

14 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1)^5*s0*s2*s1*s2, (s1*s0*s2)^2*s1*s0*(s2*s1)^4*s2> of order 6

36 facets

9 vertex figures

P/N, where N=<s1*s0*s2*s1*s0*(s2*s1)^2*(s0*s2*s1)^2*s2, s0*s1*s0*(s2*s1)^5*s0*s2*s1*s2> of order 6

36 facets

11 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 6

36 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle