Overview
- Group
- SmallGroup(1728,30243)
- Rank
- 3
- Schläfli Type
- {12,6}
- Vertices, edges, …
- 144, 432, 72
- 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
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)^2*(s2*s1*s0)^4*(s2*s1)^2> of order 2
36 facets
- 36 of {12}*24
72 vertex figures
- 72 of {6}*12
P/N, where N=<s0*s1*(s2*s1*s0)^2*(s2*s1)^2*s0*(s1*s2)^2*s1> of order 2
36 facets
- 36 of {12}*24
72 vertex figures
- 72 of {6}*12
P/N, where N=<(s1*s0)^2*(s2*s1*s0)^2*(s2*s1)^2> of order 3
24 facets
- 24 of {12}*24
48 vertex figures
- 48 of {6}*12
P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1*s0*s1*s2*s1> of order 4
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*s1*(s2*s1*s0)^2*(s2*s1)^2*s0*(s1*s2)^2*s1, (s0*s1)^2*(s2*s1)^2*s0*(s1*s2)^2*s1*s0*(s1*s2)^2> of order 4
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^5*s1*s2, s0*s1*(s2*s1*s0)^2*(s2*s1)^2*s0*(s1*s2)^2*s1> of order 4
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<(s1*s0*(s1*s2)^2)^2*s1*s0*s1*s2*s1> of order 4
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1*s0*(s1*s2)^2> of order 4
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
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)(217,325)(218,326)(219,328)(220,327)(221,329)(222,330)(223,332)(224,331)(225,333)(226,334)(227,336)(228,335)(229,349)(230,350)(231,352)(232,351)(233,353)(234,354)(235,356)(236,355)(237,357)(238,358)(239,360)(240,359)(241,337)(242,338)(243,340)(244,339)(245,341)(246,342)(247,344)(248,343)(249,345)(250,346)(251,348)(252,347)(253,397)(254,398)(255,400)(256,399)(257,401)(258,402)(259,404)(260,403)(261,405)(262,406)(263,408)(264,407)(265,421)(266,422)(267,424)(268,423)(269,425)(270,426)(271,428)(272,427)(273,429)(274,430)(275,432)(276,431)(277,409)(278,410)(279,412)(280,411)(281,413)(282,414)(283,416)(284,415)(285,417)(286,418)(287,420)(288,419)(289,361)(290,362)(291,364)(292,363)(293,365)(294,366)(295,368)(296,367)(297,369)(298,370)(299,372)(300,371)(301,385)(302,386)(303,388)(304,387)(305,389)(306,390)(307,392)(308,391)(309,393)(310,394)(311,396)(312,395)(313,373)(314,374)(315,376)(316,375)(317,377)(318,378)(319,380)(320,379)(321,381)(322,382)(323,384)(324,383);; 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)( 5, 10)( 6, 9)( 7, 11)( 8, 12)( 13, 14)( 17, 22)( 18, 21)( 19, 23)( 20, 24)( 25, 26)( 29, 34)( 30, 33)( 31, 35)( 32, 36)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 82)( 42, 81)( 43, 83)( 44, 84)( 45, 78)( 46, 77)( 47, 79)( 48, 80)( 49, 86)( 50, 85)( 51, 87)( 52, 88)( 53, 94)( 54, 93)( 55, 95)( 56, 96)( 57, 90)( 58, 89)( 59, 91)( 60, 92)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,106)( 66,105)( 67,107)( 68,108)( 69,102)( 70,101)( 71,103)( 72,104)(109,110)(113,118)(114,117)(115,119)(116,120)(121,122)(125,130)(126,129)(127,131)(128,132)(133,134)(137,142)(138,141)(139,143)(140,144)(145,182)(146,181)(147,183)(148,184)(149,190)(150,189)(151,191)(152,192)(153,186)(154,185)(155,187)(156,188)(157,194)(158,193)(159,195)(160,196)(161,202)(162,201)(163,203)(164,204)(165,198)(166,197)(167,199)(168,200)(169,206)(170,205)(171,207)(172,208)(173,214)(174,213)(175,215)(176,216)(177,210)(178,209)(179,211)(180,212)(217,218)(221,226)(222,225)(223,227)(224,228)(229,230)(233,238)(234,237)(235,239)(236,240)(241,242)(245,250)(246,249)(247,251)(248,252)(253,290)(254,289)(255,291)(256,292)(257,298)(258,297)(259,299)(260,300)(261,294)(262,293)(263,295)(264,296)(265,302)(266,301)(267,303)(268,304)(269,310)(270,309)(271,311)(272,312)(273,306)(274,305)(275,307)(276,308)(277,314)(278,313)(279,315)(280,316)(281,322)(282,321)(283,323)(284,324)(285,318)(286,317)(287,319)(288,320)(325,326)(329,334)(330,333)(331,335)(332,336)(337,338)(341,346)(342,345)(343,347)(344,348)(349,350)(353,358)(354,357)(355,359)(356,360)(361,398)(362,397)(363,399)(364,400)(365,406)(366,405)(367,407)(368,408)(369,402)(370,401)(371,403)(372,404)(373,410)(374,409)(375,411)(376,412)(377,418)(378,417)(379,419)(380,420)(381,414)(382,413)(383,415)(384,416)(385,422)(386,421)(387,423)(388,424)(389,430)(390,429)(391,431)(392,432)(393,426)(394,425)(395,427)(396,428);; 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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
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)(217,325)(218,326)(219,328)(220,327)(221,329)(222,330)(223,332)(224,331)(225,333)(226,334)(227,336)(228,335)(229,349)(230,350)(231,352)(232,351)(233,353)(234,354)(235,356)(236,355)(237,357)(238,358)(239,360)(240,359)(241,337)(242,338)(243,340)(244,339)(245,341)(246,342)(247,344)(248,343)(249,345)(250,346)(251,348)(252,347)(253,397)(254,398)(255,400)(256,399)(257,401)(258,402)(259,404)(260,403)(261,405)(262,406)(263,408)(264,407)(265,421)(266,422)(267,424)(268,423)(269,425)(270,426)(271,428)(272,427)(273,429)(274,430)(275,432)(276,431)(277,409)(278,410)(279,412)(280,411)(281,413)(282,414)(283,416)(284,415)(285,417)(286,418)(287,420)(288,419)(289,361)(290,362)(291,364)(292,363)(293,365)(294,366)(295,368)(296,367)(297,369)(298,370)(299,372)(300,371)(301,385)(302,386)(303,388)(304,387)(305,389)(306,390)(307,392)(308,391)(309,393)(310,394)(311,396)(312,395)(313,373)(314,374)(315,376)(316,375)(317,377)(318,378)(319,380)(320,379)(321,381)(322,382)(323,384)(324,383); 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)( 5, 10)( 6, 9)( 7, 11)( 8, 12)( 13, 14)( 17, 22)( 18, 21)( 19, 23)( 20, 24)( 25, 26)( 29, 34)( 30, 33)( 31, 35)( 32, 36)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 82)( 42, 81)( 43, 83)( 44, 84)( 45, 78)( 46, 77)( 47, 79)( 48, 80)( 49, 86)( 50, 85)( 51, 87)( 52, 88)( 53, 94)( 54, 93)( 55, 95)( 56, 96)( 57, 90)( 58, 89)( 59, 91)( 60, 92)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,106)( 66,105)( 67,107)( 68,108)( 69,102)( 70,101)( 71,103)( 72,104)(109,110)(113,118)(114,117)(115,119)(116,120)(121,122)(125,130)(126,129)(127,131)(128,132)(133,134)(137,142)(138,141)(139,143)(140,144)(145,182)(146,181)(147,183)(148,184)(149,190)(150,189)(151,191)(152,192)(153,186)(154,185)(155,187)(156,188)(157,194)(158,193)(159,195)(160,196)(161,202)(162,201)(163,203)(164,204)(165,198)(166,197)(167,199)(168,200)(169,206)(170,205)(171,207)(172,208)(173,214)(174,213)(175,215)(176,216)(177,210)(178,209)(179,211)(180,212)(217,218)(221,226)(222,225)(223,227)(224,228)(229,230)(233,238)(234,237)(235,239)(236,240)(241,242)(245,250)(246,249)(247,251)(248,252)(253,290)(254,289)(255,291)(256,292)(257,298)(258,297)(259,299)(260,300)(261,294)(262,293)(263,295)(264,296)(265,302)(266,301)(267,303)(268,304)(269,310)(270,309)(271,311)(272,312)(273,306)(274,305)(275,307)(276,308)(277,314)(278,313)(279,315)(280,316)(281,322)(282,321)(283,323)(284,324)(285,318)(286,317)(287,319)(288,320)(325,326)(329,334)(330,333)(331,335)(332,336)(337,338)(341,346)(342,345)(343,347)(344,348)(349,350)(353,358)(354,357)(355,359)(356,360)(361,398)(362,397)(363,399)(364,400)(365,406)(366,405)(367,407)(368,408)(369,402)(370,401)(371,403)(372,404)(373,410)(374,409)(375,411)(376,412)(377,418)(378,417)(379,419)(380,420)(381,414)(382,413)(383,415)(384,416)(385,422)(386,421)(387,423)(388,424)(389,430)(390,429)(391,431)(392,432)(393,426)(394,425)(395,427)(396,428); 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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References
None.
to this polytope.