Polytope of Type {36,24} with group SmallGroup(1728,4743)

Overview

Group: SmallGroup(1728,4743)
Rank: 3
Schläfli Type: {36,24}
Vertices, edges, …: 36, 432, 24
Order of s₀s₁s₂: 72
Order of s₀s₁s₂s₁: 2
Also known as: {36,24|2}. if this polytope has another name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

{18,6}*216a

9-fold

12-fold

18-fold

24-fold

27-fold

{4,8}*64a

36-fold

48-fold

{9,2}*36

54-fold

72-fold

108-fold

144-fold

216-fold

{2,2}*8

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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