Polytope of Type {56,8} with group SmallGroup(896,714)

Overview

Group: SmallGroup(896,714)
Rank: 3
Schläfli Type: {56,8}
Vertices, edges, …: 56, 224, 8
Order of s₀s₁s₂: 56
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

Quotients maximal quotients in bold

2-fold

4-fold

7-fold

{8,8}*128c

8-fold

14-fold

16-fold

{14,2}*56

28-fold

32-fold

{7,2}*28

56-fold

112-fold

{2,2}*8

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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