Polytope of Type {6,76} with group SmallGroup(1824,1245)

Overview

Group: SmallGroup(1824,1245)
Rank: 3
Schläfli Type: {6,76}
Vertices, edges, …: 12, 456, 152
Order of s₀s₁s₂: 114
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

2-fold

{6,76}*912b

4-fold

{6,38}*456

12-fold

{2,38}*152

19-fold

{6,4}*96

24-fold

{2,19}*76

38-fold

76-fold

152-fold

{3,2}*12

228-fold

{2,2}*8

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*s2*s1)^2> of order 2

76 facets

76 of {6}*12

8 vertex figures

4 of {76}*152
4 of {38}*76

P/N, where N=<s2*s1*s0*(s2*s1)^2*s0*(s2*s1)^15*s2> of order 2

76 facets

76 of {6}*12

6 vertex figures

6 of {76}*152

Representations

Permutation Representation (GAP)

s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 77,153)( 78,154)( 79,156)( 80,155)( 81,157)( 82,158)( 83,160)( 84,159)( 85,161)( 86,162)( 87,164)( 88,163)( 89,165)( 90,166)( 91,168)( 92,167)( 93,169)( 94,170)( 95,172)( 96,171)( 97,173)( 98,174)( 99,176)(100,175)(101,177)(102,178)(103,180)(104,179)(105,181)(106,182)(107,184)(108,183)(109,185)(110,186)(111,188)(112,187)(113,189)(114,190)(115,192)(116,191)(117,193)(118,194)(119,196)(120,195)(121,197)(122,198)(123,200)(124,199)(125,201)(126,202)(127,204)(128,203)(129,205)(130,206)(131,208)(132,207)(133,209)(134,210)(135,212)(136,211)(137,213)(138,214)(139,216)(140,215)(141,217)(142,218)(143,220)(144,219)(145,221)(146,222)(147,224)(148,223)(149,225)(150,226)(151,228)(152,227)(231,232)(235,236)(239,240)(243,244)(247,248)(251,252)(255,256)(259,260)(263,264)(267,268)(271,272)(275,276)(279,280)(283,284)(287,288)(291,292)(295,296)(299,300)(303,304)(305,381)(306,382)(307,384)(308,383)(309,385)(310,386)(311,388)(312,387)(313,389)(314,390)(315,392)(316,391)(317,393)(318,394)(319,396)(320,395)(321,397)(322,398)(323,400)(324,399)(325,401)(326,402)(327,404)(328,403)(329,405)(330,406)(331,408)(332,407)(333,409)(334,410)(335,412)(336,411)(337,413)(338,414)(339,416)(340,415)(341,417)(342,418)(343,420)(344,419)(345,421)(346,422)(347,424)(348,423)(349,425)(350,426)(351,428)(352,427)(353,429)(354,430)(355,432)(356,431)(357,433)(358,434)(359,436)(360,435)(361,437)(362,438)(363,440)(364,439)(365,441)(366,442)(367,444)(368,443)(369,445)(370,446)(371,448)(372,447)(373,449)(374,450)(375,452)(376,451)(377,453)(378,454)(379,456)(380,455);;
s1 := (  1, 77)(  2, 80)(  3, 79)(  4, 78)(  5,149)(  6,152)(  7,151)(  8,150)(  9,145)( 10,148)( 11,147)( 12,146)( 13,141)( 14,144)( 15,143)( 16,142)( 17,137)( 18,140)( 19,139)( 20,138)( 21,133)( 22,136)( 23,135)( 24,134)( 25,129)( 26,132)( 27,131)( 28,130)( 29,125)( 30,128)( 31,127)( 32,126)( 33,121)( 34,124)( 35,123)( 36,122)( 37,117)( 38,120)( 39,119)( 40,118)( 41,113)( 42,116)( 43,115)( 44,114)( 45,109)( 46,112)( 47,111)( 48,110)( 49,105)( 50,108)( 51,107)( 52,106)( 53,101)( 54,104)( 55,103)( 56,102)( 57, 97)( 58,100)( 59, 99)( 60, 98)( 61, 93)( 62, 96)( 63, 95)( 64, 94)( 65, 89)( 66, 92)( 67, 91)( 68, 90)( 69, 85)( 70, 88)( 71, 87)( 72, 86)( 73, 81)( 74, 84)( 75, 83)( 76, 82)(154,156)(157,225)(158,228)(159,227)(160,226)(161,221)(162,224)(163,223)(164,222)(165,217)(166,220)(167,219)(168,218)(169,213)(170,216)(171,215)(172,214)(173,209)(174,212)(175,211)(176,210)(177,205)(178,208)(179,207)(180,206)(181,201)(182,204)(183,203)(184,202)(185,197)(186,200)(187,199)(188,198)(189,193)(190,196)(191,195)(192,194)(229,305)(230,308)(231,307)(232,306)(233,377)(234,380)(235,379)(236,378)(237,373)(238,376)(239,375)(240,374)(241,369)(242,372)(243,371)(244,370)(245,365)(246,368)(247,367)(248,366)(249,361)(250,364)(251,363)(252,362)(253,357)(254,360)(255,359)(256,358)(257,353)(258,356)(259,355)(260,354)(261,349)(262,352)(263,351)(264,350)(265,345)(266,348)(267,347)(268,346)(269,341)(270,344)(271,343)(272,342)(273,337)(274,340)(275,339)(276,338)(277,333)(278,336)(279,335)(280,334)(281,329)(282,332)(283,331)(284,330)(285,325)(286,328)(287,327)(288,326)(289,321)(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)(297,313)(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)(382,384)(385,453)(386,456)(387,455)(388,454)(389,449)(390,452)(391,451)(392,450)(393,445)(394,448)(395,447)(396,446)(397,441)(398,444)(399,443)(400,442)(401,437)(402,440)(403,439)(404,438)(405,433)(406,436)(407,435)(408,434)(409,429)(410,432)(411,431)(412,430)(413,425)(414,428)(415,427)(416,426)(417,421)(418,424)(419,423)(420,422);;
s2 := (  1,234)(  2,233)(  3,236)(  4,235)(  5,230)(  6,229)(  7,232)(  8,231)(  9,302)( 10,301)( 11,304)( 12,303)( 13,298)( 14,297)( 15,300)( 16,299)( 17,294)( 18,293)( 19,296)( 20,295)( 21,290)( 22,289)( 23,292)( 24,291)( 25,286)( 26,285)( 27,288)( 28,287)( 29,282)( 30,281)( 31,284)( 32,283)( 33,278)( 34,277)( 35,280)( 36,279)( 37,274)( 38,273)( 39,276)( 40,275)( 41,270)( 42,269)( 43,272)( 44,271)( 45,266)( 46,265)( 47,268)( 48,267)( 49,262)( 50,261)( 51,264)( 52,263)( 53,258)( 54,257)( 55,260)( 56,259)( 57,254)( 58,253)( 59,256)( 60,255)( 61,250)( 62,249)( 63,252)( 64,251)( 65,246)( 66,245)( 67,248)( 68,247)( 69,242)( 70,241)( 71,244)( 72,243)( 73,238)( 74,237)( 75,240)( 76,239)( 77,310)( 78,309)( 79,312)( 80,311)( 81,306)( 82,305)( 83,308)( 84,307)( 85,378)( 86,377)( 87,380)( 88,379)( 89,374)( 90,373)( 91,376)( 92,375)( 93,370)( 94,369)( 95,372)( 96,371)( 97,366)( 98,365)( 99,368)(100,367)(101,362)(102,361)(103,364)(104,363)(105,358)(106,357)(107,360)(108,359)(109,354)(110,353)(111,356)(112,355)(113,350)(114,349)(115,352)(116,351)(117,346)(118,345)(119,348)(120,347)(121,342)(122,341)(123,344)(124,343)(125,338)(126,337)(127,340)(128,339)(129,334)(130,333)(131,336)(132,335)(133,330)(134,329)(135,332)(136,331)(137,326)(138,325)(139,328)(140,327)(141,322)(142,321)(143,324)(144,323)(145,318)(146,317)(147,320)(148,319)(149,314)(150,313)(151,316)(152,315)(153,386)(154,385)(155,388)(156,387)(157,382)(158,381)(159,384)(160,383)(161,454)(162,453)(163,456)(164,455)(165,450)(166,449)(167,452)(168,451)(169,446)(170,445)(171,448)(172,447)(173,442)(174,441)(175,444)(176,443)(177,438)(178,437)(179,440)(180,439)(181,434)(182,433)(183,436)(184,435)(185,430)(186,429)(187,432)(188,431)(189,426)(190,425)(191,428)(192,427)(193,422)(194,421)(195,424)(196,423)(197,418)(198,417)(199,420)(200,419)(201,414)(202,413)(203,416)(204,415)(205,410)(206,409)(207,412)(208,411)(209,406)(210,405)(211,408)(212,407)(213,402)(214,401)(215,404)(216,403)(217,398)(218,397)(219,400)(220,399)(221,394)(222,393)(223,396)(224,395)(225,390)(226,389)(227,392)(228,391);;
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, 
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(456)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 77,153)( 78,154)( 79,156)( 80,155)( 81,157)( 82,158)( 83,160)( 84,159)( 85,161)( 86,162)( 87,164)( 88,163)( 89,165)( 90,166)( 91,168)( 92,167)( 93,169)( 94,170)( 95,172)( 96,171)( 97,173)( 98,174)( 99,176)(100,175)(101,177)(102,178)(103,180)(104,179)(105,181)(106,182)(107,184)(108,183)(109,185)(110,186)(111,188)(112,187)(113,189)(114,190)(115,192)(116,191)(117,193)(118,194)(119,196)(120,195)(121,197)(122,198)(123,200)(124,199)(125,201)(126,202)(127,204)(128,203)(129,205)(130,206)(131,208)(132,207)(133,209)(134,210)(135,212)(136,211)(137,213)(138,214)(139,216)(140,215)(141,217)(142,218)(143,220)(144,219)(145,221)(146,222)(147,224)(148,223)(149,225)(150,226)(151,228)(152,227)(231,232)(235,236)(239,240)(243,244)(247,248)(251,252)(255,256)(259,260)(263,264)(267,268)(271,272)(275,276)(279,280)(283,284)(287,288)(291,292)(295,296)(299,300)(303,304)(305,381)(306,382)(307,384)(308,383)(309,385)(310,386)(311,388)(312,387)(313,389)(314,390)(315,392)(316,391)(317,393)(318,394)(319,396)(320,395)(321,397)(322,398)(323,400)(324,399)(325,401)(326,402)(327,404)(328,403)(329,405)(330,406)(331,408)(332,407)(333,409)(334,410)(335,412)(336,411)(337,413)(338,414)(339,416)(340,415)(341,417)(342,418)(343,420)(344,419)(345,421)(346,422)(347,424)(348,423)(349,425)(350,426)(351,428)(352,427)(353,429)(354,430)(355,432)(356,431)(357,433)(358,434)(359,436)(360,435)(361,437)(362,438)(363,440)(364,439)(365,441)(366,442)(367,444)(368,443)(369,445)(370,446)(371,448)(372,447)(373,449)(374,450)(375,452)(376,451)(377,453)(378,454)(379,456)(380,455);
s1 := Sym(456)!(  1, 77)(  2, 80)(  3, 79)(  4, 78)(  5,149)(  6,152)(  7,151)(  8,150)(  9,145)( 10,148)( 11,147)( 12,146)( 13,141)( 14,144)( 15,143)( 16,142)( 17,137)( 18,140)( 19,139)( 20,138)( 21,133)( 22,136)( 23,135)( 24,134)( 25,129)( 26,132)( 27,131)( 28,130)( 29,125)( 30,128)( 31,127)( 32,126)( 33,121)( 34,124)( 35,123)( 36,122)( 37,117)( 38,120)( 39,119)( 40,118)( 41,113)( 42,116)( 43,115)( 44,114)( 45,109)( 46,112)( 47,111)( 48,110)( 49,105)( 50,108)( 51,107)( 52,106)( 53,101)( 54,104)( 55,103)( 56,102)( 57, 97)( 58,100)( 59, 99)( 60, 98)( 61, 93)( 62, 96)( 63, 95)( 64, 94)( 65, 89)( 66, 92)( 67, 91)( 68, 90)( 69, 85)( 70, 88)( 71, 87)( 72, 86)( 73, 81)( 74, 84)( 75, 83)( 76, 82)(154,156)(157,225)(158,228)(159,227)(160,226)(161,221)(162,224)(163,223)(164,222)(165,217)(166,220)(167,219)(168,218)(169,213)(170,216)(171,215)(172,214)(173,209)(174,212)(175,211)(176,210)(177,205)(178,208)(179,207)(180,206)(181,201)(182,204)(183,203)(184,202)(185,197)(186,200)(187,199)(188,198)(189,193)(190,196)(191,195)(192,194)(229,305)(230,308)(231,307)(232,306)(233,377)(234,380)(235,379)(236,378)(237,373)(238,376)(239,375)(240,374)(241,369)(242,372)(243,371)(244,370)(245,365)(246,368)(247,367)(248,366)(249,361)(250,364)(251,363)(252,362)(253,357)(254,360)(255,359)(256,358)(257,353)(258,356)(259,355)(260,354)(261,349)(262,352)(263,351)(264,350)(265,345)(266,348)(267,347)(268,346)(269,341)(270,344)(271,343)(272,342)(273,337)(274,340)(275,339)(276,338)(277,333)(278,336)(279,335)(280,334)(281,329)(282,332)(283,331)(284,330)(285,325)(286,328)(287,327)(288,326)(289,321)(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)(297,313)(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)(382,384)(385,453)(386,456)(387,455)(388,454)(389,449)(390,452)(391,451)(392,450)(393,445)(394,448)(395,447)(396,446)(397,441)(398,444)(399,443)(400,442)(401,437)(402,440)(403,439)(404,438)(405,433)(406,436)(407,435)(408,434)(409,429)(410,432)(411,431)(412,430)(413,425)(414,428)(415,427)(416,426)(417,421)(418,424)(419,423)(420,422);
s2 := Sym(456)!(  1,234)(  2,233)(  3,236)(  4,235)(  5,230)(  6,229)(  7,232)(  8,231)(  9,302)( 10,301)( 11,304)( 12,303)( 13,298)( 14,297)( 15,300)( 16,299)( 17,294)( 18,293)( 19,296)( 20,295)( 21,290)( 22,289)( 23,292)( 24,291)( 25,286)( 26,285)( 27,288)( 28,287)( 29,282)( 30,281)( 31,284)( 32,283)( 33,278)( 34,277)( 35,280)( 36,279)( 37,274)( 38,273)( 39,276)( 40,275)( 41,270)( 42,269)( 43,272)( 44,271)( 45,266)( 46,265)( 47,268)( 48,267)( 49,262)( 50,261)( 51,264)( 52,263)( 53,258)( 54,257)( 55,260)( 56,259)( 57,254)( 58,253)( 59,256)( 60,255)( 61,250)( 62,249)( 63,252)( 64,251)( 65,246)( 66,245)( 67,248)( 68,247)( 69,242)( 70,241)( 71,244)( 72,243)( 73,238)( 74,237)( 75,240)( 76,239)( 77,310)( 78,309)( 79,312)( 80,311)( 81,306)( 82,305)( 83,308)( 84,307)( 85,378)( 86,377)( 87,380)( 88,379)( 89,374)( 90,373)( 91,376)( 92,375)( 93,370)( 94,369)( 95,372)( 96,371)( 97,366)( 98,365)( 99,368)(100,367)(101,362)(102,361)(103,364)(104,363)(105,358)(106,357)(107,360)(108,359)(109,354)(110,353)(111,356)(112,355)(113,350)(114,349)(115,352)(116,351)(117,346)(118,345)(119,348)(120,347)(121,342)(122,341)(123,344)(124,343)(125,338)(126,337)(127,340)(128,339)(129,334)(130,333)(131,336)(132,335)(133,330)(134,329)(135,332)(136,331)(137,326)(138,325)(139,328)(140,327)(141,322)(142,321)(143,324)(144,323)(145,318)(146,317)(147,320)(148,319)(149,314)(150,313)(151,316)(152,315)(153,386)(154,385)(155,388)(156,387)(157,382)(158,381)(159,384)(160,383)(161,454)(162,453)(163,456)(164,455)(165,450)(166,449)(167,452)(168,451)(169,446)(170,445)(171,448)(172,447)(173,442)(174,441)(175,444)(176,443)(177,438)(178,437)(179,440)(180,439)(181,434)(182,433)(183,436)(184,435)(185,430)(186,429)(187,432)(188,431)(189,426)(190,425)(191,428)(192,427)(193,422)(194,421)(195,424)(196,423)(197,418)(198,417)(199,420)(200,419)(201,414)(202,413)(203,416)(204,415)(205,410)(206,409)(207,412)(208,411)(209,406)(210,405)(211,408)(212,407)(213,402)(214,401)(215,404)(216,403)(217,398)(218,397)(219,400)(220,399)(221,394)(222,393)(223,396)(224,395)(225,390)(226,389)(227,392)(228,391);
poly := sub<Sym(456)|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, 
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1 >;

References

None.

to this polytope.