Part of the Atlas of Small Regular Polytopes

Polytope of Type {36,6}

Atlas Canonical Name {36,6}*1728a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,30174)
Rank
3
Schläfli Type
{36,6}
Vertices, edges, …
144, 432, 24
Order of s0s1s2
36
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

8-fold

9-fold

12-fold

16-fold

18-fold

24-fold

36-fold

48-fold

72-fold

108-fold

144-fold

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

12 facets

72 vertex figures

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

12 facets

72 vertex figures

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

8 facets

72 vertex figures

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

6 facets

36 vertex figures

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

6 facets

36 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, (s0*s1)^14*(s0*s2*s1)^3*s0*s1*s2> of order 4

6 facets

36 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, (s0*s1)^14*(s0*s2*s1)^3*s0*s1> of order 4

6 facets

36 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)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 49,105)( 50,106)( 51,108)( 52,107)( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)( 59,100)( 60, 99)( 61, 93)( 62, 94)( 63, 96)( 64, 95)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 85)( 70, 86)( 71, 88)( 72, 87)(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)(217,325)(218,326)(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)(226,330)(227,332)(228,331)(229,357)(230,358)(231,360)(232,359)(233,353)(234,354)(235,356)(236,355)(237,349)(238,350)(239,352)(240,351)(241,345)(242,346)(243,348)(244,347)(245,341)(246,342)(247,344)(248,343)(249,337)(250,338)(251,340)(252,339)(253,397)(254,398)(255,400)(256,399)(257,405)(258,406)(259,408)(260,407)(261,401)(262,402)(263,404)(264,403)(265,429)(266,430)(267,432)(268,431)(269,425)(270,426)(271,428)(272,427)(273,421)(274,422)(275,424)(276,423)(277,417)(278,418)(279,420)(280,419)(281,413)(282,414)(283,416)(284,415)(285,409)(286,410)(287,412)(288,411)(289,361)(290,362)(291,364)(292,363)(293,369)(294,370)(295,372)(296,371)(297,365)(298,366)(299,368)(300,367)(301,393)(302,394)(303,396)(304,395)(305,389)(306,390)(307,392)(308,391)(309,385)(310,386)(311,388)(312,387)(313,381)(314,382)(315,384)(316,383)(317,377)(318,378)(319,380)(320,379)(321,373)(322,374)(323,376)(324,375);;
s1 := (  1,265)(  2,268)(  3,267)(  4,266)(  5,273)(  6,276)(  7,275)(  8,274)(  9,269)( 10,272)( 11,271)( 12,270)( 13,253)( 14,256)( 15,255)( 16,254)( 17,261)( 18,264)( 19,263)( 20,262)( 21,257)( 22,260)( 23,259)( 24,258)( 25,285)( 26,288)( 27,287)( 28,286)( 29,281)( 30,284)( 31,283)( 32,282)( 33,277)( 34,280)( 35,279)( 36,278)( 37,229)( 38,232)( 39,231)( 40,230)( 41,237)( 42,240)( 43,239)( 44,238)( 45,233)( 46,236)( 47,235)( 48,234)( 49,217)( 50,220)( 51,219)( 52,218)( 53,225)( 54,228)( 55,227)( 56,226)( 57,221)( 58,224)( 59,223)( 60,222)( 61,249)( 62,252)( 63,251)( 64,250)( 65,245)( 66,248)( 67,247)( 68,246)( 69,241)( 70,244)( 71,243)( 72,242)( 73,301)( 74,304)( 75,303)( 76,302)( 77,309)( 78,312)( 79,311)( 80,310)( 81,305)( 82,308)( 83,307)( 84,306)( 85,289)( 86,292)( 87,291)( 88,290)( 89,297)( 90,300)( 91,299)( 92,298)( 93,293)( 94,296)( 95,295)( 96,294)( 97,321)( 98,324)( 99,323)(100,322)(101,317)(102,320)(103,319)(104,318)(105,313)(106,316)(107,315)(108,314)(109,373)(110,376)(111,375)(112,374)(113,381)(114,384)(115,383)(116,382)(117,377)(118,380)(119,379)(120,378)(121,361)(122,364)(123,363)(124,362)(125,369)(126,372)(127,371)(128,370)(129,365)(130,368)(131,367)(132,366)(133,393)(134,396)(135,395)(136,394)(137,389)(138,392)(139,391)(140,390)(141,385)(142,388)(143,387)(144,386)(145,337)(146,340)(147,339)(148,338)(149,345)(150,348)(151,347)(152,346)(153,341)(154,344)(155,343)(156,342)(157,325)(158,328)(159,327)(160,326)(161,333)(162,336)(163,335)(164,334)(165,329)(166,332)(167,331)(168,330)(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)(176,354)(177,349)(178,352)(179,351)(180,350)(181,409)(182,412)(183,411)(184,410)(185,417)(186,420)(187,419)(188,418)(189,413)(190,416)(191,415)(192,414)(193,397)(194,400)(195,399)(196,398)(197,405)(198,408)(199,407)(200,406)(201,401)(202,404)(203,403)(204,402)(205,429)(206,432)(207,431)(208,430)(209,425)(210,428)(211,427)(212,426)(213,421)(214,424)(215,423)(216,422);;
s2 := (  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)( 29, 30)( 33, 34)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 79)( 44, 80)( 45, 82)( 46, 81)( 47, 83)( 48, 84)( 49, 86)( 50, 85)( 51, 87)( 52, 88)( 53, 90)( 54, 89)( 55, 91)( 56, 92)( 57, 94)( 58, 93)( 59, 95)( 60, 96)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,102)( 66,101)( 67,103)( 68,104)( 69,106)( 70,105)( 71,107)( 72,108)(109,110)(113,114)(117,118)(121,122)(125,126)(129,130)(133,134)(137,138)(141,142)(145,182)(146,181)(147,183)(148,184)(149,186)(150,185)(151,187)(152,188)(153,190)(154,189)(155,191)(156,192)(157,194)(158,193)(159,195)(160,196)(161,198)(162,197)(163,199)(164,200)(165,202)(166,201)(167,203)(168,204)(169,206)(170,205)(171,207)(172,208)(173,210)(174,209)(175,211)(176,212)(177,214)(178,213)(179,215)(180,216)(217,218)(221,222)(225,226)(229,230)(233,234)(237,238)(241,242)(245,246)(249,250)(253,290)(254,289)(255,291)(256,292)(257,294)(258,293)(259,295)(260,296)(261,298)(262,297)(263,299)(264,300)(265,302)(266,301)(267,303)(268,304)(269,306)(270,305)(271,307)(272,308)(273,310)(274,309)(275,311)(276,312)(277,314)(278,313)(279,315)(280,316)(281,318)(282,317)(283,319)(284,320)(285,322)(286,321)(287,323)(288,324)(325,326)(329,330)(333,334)(337,338)(341,342)(345,346)(349,350)(353,354)(357,358)(361,398)(362,397)(363,399)(364,400)(365,402)(366,401)(367,403)(368,404)(369,406)(370,405)(371,407)(372,408)(373,410)(374,409)(375,411)(376,412)(377,414)(378,413)(379,415)(380,416)(381,418)(382,417)(383,419)(384,420)(385,422)(386,421)(387,423)(388,424)(389,426)(390,425)(391,427)(392,428)(393,430)(394,429)(395,431)(396,432);;
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*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*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 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(432)!(  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)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 49,105)( 50,106)( 51,108)( 52,107)( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)( 59,100)( 60, 99)( 61, 93)( 62, 94)( 63, 96)( 64, 95)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 85)( 70, 86)( 71, 88)( 72, 87)(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)(217,325)(218,326)(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)(226,330)(227,332)(228,331)(229,357)(230,358)(231,360)(232,359)(233,353)(234,354)(235,356)(236,355)(237,349)(238,350)(239,352)(240,351)(241,345)(242,346)(243,348)(244,347)(245,341)(246,342)(247,344)(248,343)(249,337)(250,338)(251,340)(252,339)(253,397)(254,398)(255,400)(256,399)(257,405)(258,406)(259,408)(260,407)(261,401)(262,402)(263,404)(264,403)(265,429)(266,430)(267,432)(268,431)(269,425)(270,426)(271,428)(272,427)(273,421)(274,422)(275,424)(276,423)(277,417)(278,418)(279,420)(280,419)(281,413)(282,414)(283,416)(284,415)(285,409)(286,410)(287,412)(288,411)(289,361)(290,362)(291,364)(292,363)(293,369)(294,370)(295,372)(296,371)(297,365)(298,366)(299,368)(300,367)(301,393)(302,394)(303,396)(304,395)(305,389)(306,390)(307,392)(308,391)(309,385)(310,386)(311,388)(312,387)(313,381)(314,382)(315,384)(316,383)(317,377)(318,378)(319,380)(320,379)(321,373)(322,374)(323,376)(324,375);
s1 := Sym(432)!(  1,265)(  2,268)(  3,267)(  4,266)(  5,273)(  6,276)(  7,275)(  8,274)(  9,269)( 10,272)( 11,271)( 12,270)( 13,253)( 14,256)( 15,255)( 16,254)( 17,261)( 18,264)( 19,263)( 20,262)( 21,257)( 22,260)( 23,259)( 24,258)( 25,285)( 26,288)( 27,287)( 28,286)( 29,281)( 30,284)( 31,283)( 32,282)( 33,277)( 34,280)( 35,279)( 36,278)( 37,229)( 38,232)( 39,231)( 40,230)( 41,237)( 42,240)( 43,239)( 44,238)( 45,233)( 46,236)( 47,235)( 48,234)( 49,217)( 50,220)( 51,219)( 52,218)( 53,225)( 54,228)( 55,227)( 56,226)( 57,221)( 58,224)( 59,223)( 60,222)( 61,249)( 62,252)( 63,251)( 64,250)( 65,245)( 66,248)( 67,247)( 68,246)( 69,241)( 70,244)( 71,243)( 72,242)( 73,301)( 74,304)( 75,303)( 76,302)( 77,309)( 78,312)( 79,311)( 80,310)( 81,305)( 82,308)( 83,307)( 84,306)( 85,289)( 86,292)( 87,291)( 88,290)( 89,297)( 90,300)( 91,299)( 92,298)( 93,293)( 94,296)( 95,295)( 96,294)( 97,321)( 98,324)( 99,323)(100,322)(101,317)(102,320)(103,319)(104,318)(105,313)(106,316)(107,315)(108,314)(109,373)(110,376)(111,375)(112,374)(113,381)(114,384)(115,383)(116,382)(117,377)(118,380)(119,379)(120,378)(121,361)(122,364)(123,363)(124,362)(125,369)(126,372)(127,371)(128,370)(129,365)(130,368)(131,367)(132,366)(133,393)(134,396)(135,395)(136,394)(137,389)(138,392)(139,391)(140,390)(141,385)(142,388)(143,387)(144,386)(145,337)(146,340)(147,339)(148,338)(149,345)(150,348)(151,347)(152,346)(153,341)(154,344)(155,343)(156,342)(157,325)(158,328)(159,327)(160,326)(161,333)(162,336)(163,335)(164,334)(165,329)(166,332)(167,331)(168,330)(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)(176,354)(177,349)(178,352)(179,351)(180,350)(181,409)(182,412)(183,411)(184,410)(185,417)(186,420)(187,419)(188,418)(189,413)(190,416)(191,415)(192,414)(193,397)(194,400)(195,399)(196,398)(197,405)(198,408)(199,407)(200,406)(201,401)(202,404)(203,403)(204,402)(205,429)(206,432)(207,431)(208,430)(209,425)(210,428)(211,427)(212,426)(213,421)(214,424)(215,423)(216,422);
s2 := Sym(432)!(  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)( 29, 30)( 33, 34)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 79)( 44, 80)( 45, 82)( 46, 81)( 47, 83)( 48, 84)( 49, 86)( 50, 85)( 51, 87)( 52, 88)( 53, 90)( 54, 89)( 55, 91)( 56, 92)( 57, 94)( 58, 93)( 59, 95)( 60, 96)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,102)( 66,101)( 67,103)( 68,104)( 69,106)( 70,105)( 71,107)( 72,108)(109,110)(113,114)(117,118)(121,122)(125,126)(129,130)(133,134)(137,138)(141,142)(145,182)(146,181)(147,183)(148,184)(149,186)(150,185)(151,187)(152,188)(153,190)(154,189)(155,191)(156,192)(157,194)(158,193)(159,195)(160,196)(161,198)(162,197)(163,199)(164,200)(165,202)(166,201)(167,203)(168,204)(169,206)(170,205)(171,207)(172,208)(173,210)(174,209)(175,211)(176,212)(177,214)(178,213)(179,215)(180,216)(217,218)(221,222)(225,226)(229,230)(233,234)(237,238)(241,242)(245,246)(249,250)(253,290)(254,289)(255,291)(256,292)(257,294)(258,293)(259,295)(260,296)(261,298)(262,297)(263,299)(264,300)(265,302)(266,301)(267,303)(268,304)(269,306)(270,305)(271,307)(272,308)(273,310)(274,309)(275,311)(276,312)(277,314)(278,313)(279,315)(280,316)(281,318)(282,317)(283,319)(284,320)(285,322)(286,321)(287,323)(288,324)(325,326)(329,330)(333,334)(337,338)(341,342)(345,346)(349,350)(353,354)(357,358)(361,398)(362,397)(363,399)(364,400)(365,402)(366,401)(367,403)(368,404)(369,406)(370,405)(371,407)(372,408)(373,410)(374,409)(375,411)(376,412)(377,414)(378,413)(379,415)(380,416)(381,418)(382,417)(383,419)(384,420)(385,422)(386,421)(387,423)(388,424)(389,426)(390,425)(391,427)(392,428)(393,430)(394,429)(395,431)(396,432);
poly := sub<Sym(432)|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*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*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 >;

References

None.

to this polytope.

Twisty Puzzle