Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*1728f

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,30326)
Rank
3
Schläfli Type
{6,12}
Vertices, edges, …
72, 432, 144
Order of s0s1s2
12
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

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=<s0*s1*s0*(s2*s1)^4*s0*s2*s1*s0*(s1*s2)^2> of order 2

72 facets

36 vertex figures

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

72 facets

36 vertex figures

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

72 facets

48 vertex figures

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

108 facets

36 vertex figures

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

48 facets

40 vertex figures

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

54 facets

24 vertex figures

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

36 facets

18 vertex figures

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

36 facets

30 vertex figures

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

24 facets

20 vertex figures

P/N, where N=<s1*s0*(s1*s2)^3*s1*s0*s2*s1, s0*s1*s2*s1*s0*(s2*s1)^4*s2> of order 6

24 facets

24 vertex figures

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 6

24 facets

32 vertex figures

P/N, where N=<(s0*s1)^3, s0*s1*s0*(s2*s1)^3*s0*s2*s1> of order 6

36 facets

20 vertex figures

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

24 facets

20 vertex figures

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

24 facets

20 vertex figures

P/N, where N=<(s1*s2)^2, s0*(s1*s2)^5*s1*s0*s2> of order 12

12 facets

18 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s1*s2)^6> of order 12

12 facets

14 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*s2*s1*s0*(s1*s2)^5> of order 12

12 facets

10 vertex figures

P/N, where N=<(s0*s1)^3, s0*(s1*s0*s2)^2*s1> of order 12

18 facets

16 vertex figures

P/N, where N=<(s0*s1)^3, s0*s1*s0*(s2*s1)^3*s0*s2*s1, s0*s1*s2*s1*s0*(s2*s1)^4*s2> of order 12

18 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle