Polytope of Type {56,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {56,8}*896b
Also Known As : {56,8|2}. if this polytope has another name.
Group : SmallGroup(896,714)
Rank : 3
Schlafli Type : {56,8}
Number of vertices, edges, etc : 56, 224, 8
Order of s0s1s2 : 56
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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}*448a, {28,8}*448a
4-fold quotients : {28,4}*224, {56,2}*224, {14,8}*224
7-fold quotients : {8,8}*128b
8-fold quotients : {28,2}*112, {14,4}*112
14-fold quotients : {4,8}*64a, {8,4}*64a
16-fold quotients : {14,2}*56
28-fold quotients : {4,4}*32, {2,8}*32, {8,2}*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, {112,8}*1792d, {56,16}*1792d, {112,8}*1792f, {56,16}*1792f
Permutation Representation (GAP) :
s0 := ( 1,337)( 2,343)( 3,342)( 4,341)( 5,340)( 6,339)( 7,338)( 8,344)
( 9,350)( 10,349)( 11,348)( 12,347)( 13,346)( 14,345)( 15,351)( 16,357)
( 17,356)( 18,355)( 19,354)( 20,353)( 21,352)( 22,358)( 23,364)( 24,363)
( 25,362)( 26,361)( 27,360)( 28,359)( 29,365)( 30,371)( 31,370)( 32,369)
( 33,368)( 34,367)( 35,366)( 36,372)( 37,378)( 38,377)( 39,376)( 40,375)
( 41,374)( 42,373)( 43,379)( 44,385)( 45,384)( 46,383)( 47,382)( 48,381)
( 49,380)( 50,386)( 51,392)( 52,391)( 53,390)( 54,389)( 55,388)( 56,387)
( 57,414)( 58,420)( 59,419)( 60,418)( 61,417)( 62,416)( 63,415)( 64,407)
( 65,413)( 66,412)( 67,411)( 68,410)( 69,409)( 70,408)( 71,400)( 72,406)
( 73,405)( 74,404)( 75,403)( 76,402)( 77,401)( 78,393)( 79,399)( 80,398)
( 81,397)( 82,396)( 83,395)( 84,394)( 85,442)( 86,448)( 87,447)( 88,446)
( 89,445)( 90,444)( 91,443)( 92,435)( 93,441)( 94,440)( 95,439)( 96,438)
( 97,437)( 98,436)( 99,428)(100,434)(101,433)(102,432)(103,431)(104,430)
(105,429)(106,421)(107,427)(108,426)(109,425)(110,424)(111,423)(112,422)
(113,225)(114,231)(115,230)(116,229)(117,228)(118,227)(119,226)(120,232)
(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,239)(128,245)
(129,244)(130,243)(131,242)(132,241)(133,240)(134,246)(135,252)(136,251)
(137,250)(138,249)(139,248)(140,247)(141,253)(142,259)(143,258)(144,257)
(145,256)(146,255)(147,254)(148,260)(149,266)(150,265)(151,264)(152,263)
(153,262)(154,261)(155,267)(156,273)(157,272)(158,271)(159,270)(160,269)
(161,268)(162,274)(163,280)(164,279)(165,278)(166,277)(167,276)(168,275)
(169,302)(170,308)(171,307)(172,306)(173,305)(174,304)(175,303)(176,295)
(177,301)(178,300)(179,299)(180,298)(181,297)(182,296)(183,288)(184,294)
(185,293)(186,292)(187,291)(188,290)(189,289)(190,281)(191,287)(192,286)
(193,285)(194,284)(195,283)(196,282)(197,330)(198,336)(199,335)(200,334)
(201,333)(202,332)(203,331)(204,323)(205,329)(206,328)(207,327)(208,326)
(209,325)(210,324)(211,316)(212,322)(213,321)(214,320)(215,319)(216,318)
(217,317)(218,309)(219,315)(220,314)(221,313)(222,312)(223,311)(224,310);;
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, s0*s1*s2*s1*s0*s1*s2*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*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)!( 1,337)( 2,343)( 3,342)( 4,341)( 5,340)( 6,339)( 7,338)
( 8,344)( 9,350)( 10,349)( 11,348)( 12,347)( 13,346)( 14,345)( 15,351)
( 16,357)( 17,356)( 18,355)( 19,354)( 20,353)( 21,352)( 22,358)( 23,364)
( 24,363)( 25,362)( 26,361)( 27,360)( 28,359)( 29,365)( 30,371)( 31,370)
( 32,369)( 33,368)( 34,367)( 35,366)( 36,372)( 37,378)( 38,377)( 39,376)
( 40,375)( 41,374)( 42,373)( 43,379)( 44,385)( 45,384)( 46,383)( 47,382)
( 48,381)( 49,380)( 50,386)( 51,392)( 52,391)( 53,390)( 54,389)( 55,388)
( 56,387)( 57,414)( 58,420)( 59,419)( 60,418)( 61,417)( 62,416)( 63,415)
( 64,407)( 65,413)( 66,412)( 67,411)( 68,410)( 69,409)( 70,408)( 71,400)
( 72,406)( 73,405)( 74,404)( 75,403)( 76,402)( 77,401)( 78,393)( 79,399)
( 80,398)( 81,397)( 82,396)( 83,395)( 84,394)( 85,442)( 86,448)( 87,447)
( 88,446)( 89,445)( 90,444)( 91,443)( 92,435)( 93,441)( 94,440)( 95,439)
( 96,438)( 97,437)( 98,436)( 99,428)(100,434)(101,433)(102,432)(103,431)
(104,430)(105,429)(106,421)(107,427)(108,426)(109,425)(110,424)(111,423)
(112,422)(113,225)(114,231)(115,230)(116,229)(117,228)(118,227)(119,226)
(120,232)(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,239)
(128,245)(129,244)(130,243)(131,242)(132,241)(133,240)(134,246)(135,252)
(136,251)(137,250)(138,249)(139,248)(140,247)(141,253)(142,259)(143,258)
(144,257)(145,256)(146,255)(147,254)(148,260)(149,266)(150,265)(151,264)
(152,263)(153,262)(154,261)(155,267)(156,273)(157,272)(158,271)(159,270)
(160,269)(161,268)(162,274)(163,280)(164,279)(165,278)(166,277)(167,276)
(168,275)(169,302)(170,308)(171,307)(172,306)(173,305)(174,304)(175,303)
(176,295)(177,301)(178,300)(179,299)(180,298)(181,297)(182,296)(183,288)
(184,294)(185,293)(186,292)(187,291)(188,290)(189,289)(190,281)(191,287)
(192,286)(193,285)(194,284)(195,283)(196,282)(197,330)(198,336)(199,335)
(200,334)(201,333)(202,332)(203,331)(204,323)(205,329)(206,328)(207,327)
(208,326)(209,325)(210,324)(211,316)(212,322)(213,321)(214,320)(215,319)
(216,318)(217,317)(218,309)(219,315)(220,314)(221,313)(222,312)(223,311)
(224,310);
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, s0*s1*s2*s1*s0*s1*s2*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*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 : None.
to this polytope