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