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Polytope of Type {56,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {56,8}*896c
if this polytope has a name.
Group : SmallGroup(896,870)
Rank : 3
Schlafli Type : {56,8}
Number of vertices, edges, etc : 56, 224, 8
Order of s0s1s2 : 56
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {56,8,2} of size 1792
Vertex Figure Of :
   {2,56,8} of size 1792
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {56,4}*448b, {28,8}*448b
   4-fold quotients : {28,4}*224
   7-fold quotients : {8,8}*128d
   8-fold quotients : {28,2}*112, {14,4}*112
   14-fold quotients : {4,8}*64b, {8,4}*64b
   16-fold quotients : {14,2}*56
   28-fold quotients : {4,4}*32
   32-fold quotients : {7,2}*28
   56-fold quotients : {2,4}*16, {4,2}*16
   112-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {56,8}*1792a
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, 51)( 30, 50)( 31, 56)( 32, 55)
( 33, 54)( 34, 53)( 35, 52)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)
( 41, 46)( 42, 45)( 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, 93)( 86, 92)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)( 99,107)
(100,106)(101,112)(102,111)(103,110)(104,109)(105,108)(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)(253,331)(254,330)(255,336)(256,335)(257,334)(258,333)
(259,332)(260,324)(261,323)(262,329)(263,328)(264,327)(265,326)(266,325)
(267,317)(268,316)(269,322)(270,321)(271,320)(272,319)(273,318)(274,310)
(275,309)(276,315)(277,314)(278,313)(279,312)(280,311)(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)( 29,162)( 30,163)( 31,164)( 32,165)
( 33,166)( 34,167)( 35,168)( 36,155)( 37,156)( 38,157)( 39,158)( 40,159)
( 41,160)( 42,161)( 43,148)( 44,149)( 45,150)( 46,151)( 47,152)( 48,153)
( 49,154)( 50,141)( 51,142)( 52,143)( 53,144)( 54,145)( 55,146)( 56,147)
( 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)( 85,211)( 86,212)( 87,213)( 88,214)
( 89,215)( 90,216)( 91,217)( 92,218)( 93,219)( 94,220)( 95,221)( 96,222)
( 97,223)( 98,224)( 99,197)(100,198)(101,199)(102,200)(103,201)(104,202)
(105,203)(106,204)(107,205)(108,206)(109,207)(110,208)(111,209)(112,210)
(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)(253,386)(254,387)(255,388)(256,389)
(257,390)(258,391)(259,392)(260,379)(261,380)(262,381)(263,382)(264,383)
(265,384)(266,385)(267,372)(268,373)(269,374)(270,375)(271,376)(272,377)
(273,378)(274,365)(275,366)(276,367)(277,368)(278,369)(279,370)(280,371)
(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)(309,435)(310,436)(311,437)(312,438)
(313,439)(314,440)(315,441)(316,442)(317,443)(318,444)(319,445)(320,446)
(321,447)(322,448)(323,421)(324,422)(325,423)(326,424)(327,425)(328,426)
(329,427)(330,428)(331,429)(332,430)(333,431)(334,432)(335,433)(336,434);;
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*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
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*s2*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s0*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 ];;
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, 51)( 30, 50)( 31, 56)
( 32, 55)( 33, 54)( 34, 53)( 35, 52)( 36, 44)( 37, 43)( 38, 49)( 39, 48)
( 40, 47)( 41, 46)( 42, 45)( 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, 93)( 86, 92)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)
( 99,107)(100,106)(101,112)(102,111)(103,110)(104,109)(105,108)(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)(253,331)(254,330)(255,336)(256,335)(257,334)
(258,333)(259,332)(260,324)(261,323)(262,329)(263,328)(264,327)(265,326)
(266,325)(267,317)(268,316)(269,322)(270,321)(271,320)(272,319)(273,318)
(274,310)(275,309)(276,315)(277,314)(278,313)(279,312)(280,311)(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)( 29,162)( 30,163)( 31,164)
( 32,165)( 33,166)( 34,167)( 35,168)( 36,155)( 37,156)( 38,157)( 39,158)
( 40,159)( 41,160)( 42,161)( 43,148)( 44,149)( 45,150)( 46,151)( 47,152)
( 48,153)( 49,154)( 50,141)( 51,142)( 52,143)( 53,144)( 54,145)( 55,146)
( 56,147)( 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)( 85,211)( 86,212)( 87,213)
( 88,214)( 89,215)( 90,216)( 91,217)( 92,218)( 93,219)( 94,220)( 95,221)
( 96,222)( 97,223)( 98,224)( 99,197)(100,198)(101,199)(102,200)(103,201)
(104,202)(105,203)(106,204)(107,205)(108,206)(109,207)(110,208)(111,209)
(112,210)(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)(253,386)(254,387)(255,388)
(256,389)(257,390)(258,391)(259,392)(260,379)(261,380)(262,381)(263,382)
(264,383)(265,384)(266,385)(267,372)(268,373)(269,374)(270,375)(271,376)
(272,377)(273,378)(274,365)(275,366)(276,367)(277,368)(278,369)(279,370)
(280,371)(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)(309,435)(310,436)(311,437)
(312,438)(313,439)(314,440)(315,441)(316,442)(317,443)(318,444)(319,445)
(320,446)(321,447)(322,448)(323,421)(324,422)(325,423)(326,424)(327,425)
(328,426)(329,427)(330,428)(331,429)(332,430)(333,431)(334,432)(335,433)
(336,434);
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*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
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*s2*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s0*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 >; 
 
References : None.
to this polytope