Polytope of Type {2,56,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,56,8}*1792c
if this polytope has a name.
Group : SmallGroup(1792,145174)
Rank : 4
Schlafli Type : {2,56,8}
Number of vertices, edges, etc : 2, 56, 224, 8
Order of s0s1s2s3 : 56
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,56,4}*896a, {2,28,8}*896a
4-fold quotients : {2,28,4}*448, {2,56,2}*448, {2,14,8}*448
7-fold quotients : {2,8,8}*256b
8-fold quotients : {2,28,2}*224, {2,14,4}*224
14-fold quotients : {2,4,8}*128a, {2,8,4}*128a
16-fold quotients : {2,14,2}*112
28-fold quotients : {2,4,4}*64, {2,2,8}*64, {2,8,2}*64
32-fold quotients : {2,7,2}*56
56-fold quotients : {2,2,4}*32, {2,4,2}*32
112-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,339)( 4,345)( 5,344)( 6,343)( 7,342)( 8,341)( 9,340)( 10,346)
( 11,352)( 12,351)( 13,350)( 14,349)( 15,348)( 16,347)( 17,353)( 18,359)
( 19,358)( 20,357)( 21,356)( 22,355)( 23,354)( 24,360)( 25,366)( 26,365)
( 27,364)( 28,363)( 29,362)( 30,361)( 31,367)( 32,373)( 33,372)( 34,371)
( 35,370)( 36,369)( 37,368)( 38,374)( 39,380)( 40,379)( 41,378)( 42,377)
( 43,376)( 44,375)( 45,381)( 46,387)( 47,386)( 48,385)( 49,384)( 50,383)
( 51,382)( 52,388)( 53,394)( 54,393)( 55,392)( 56,391)( 57,390)( 58,389)
( 59,416)( 60,422)( 61,421)( 62,420)( 63,419)( 64,418)( 65,417)( 66,409)
( 67,415)( 68,414)( 69,413)( 70,412)( 71,411)( 72,410)( 73,402)( 74,408)
( 75,407)( 76,406)( 77,405)( 78,404)( 79,403)( 80,395)( 81,401)( 82,400)
( 83,399)( 84,398)( 85,397)( 86,396)( 87,444)( 88,450)( 89,449)( 90,448)
( 91,447)( 92,446)( 93,445)( 94,437)( 95,443)( 96,442)( 97,441)( 98,440)
( 99,439)(100,438)(101,430)(102,436)(103,435)(104,434)(105,433)(106,432)
(107,431)(108,423)(109,429)(110,428)(111,427)(112,426)(113,425)(114,424)
(115,227)(116,233)(117,232)(118,231)(119,230)(120,229)(121,228)(122,234)
(123,240)(124,239)(125,238)(126,237)(127,236)(128,235)(129,241)(130,247)
(131,246)(132,245)(133,244)(134,243)(135,242)(136,248)(137,254)(138,253)
(139,252)(140,251)(141,250)(142,249)(143,255)(144,261)(145,260)(146,259)
(147,258)(148,257)(149,256)(150,262)(151,268)(152,267)(153,266)(154,265)
(155,264)(156,263)(157,269)(158,275)(159,274)(160,273)(161,272)(162,271)
(163,270)(164,276)(165,282)(166,281)(167,280)(168,279)(169,278)(170,277)
(171,304)(172,310)(173,309)(174,308)(175,307)(176,306)(177,305)(178,297)
(179,303)(180,302)(181,301)(182,300)(183,299)(184,298)(185,290)(186,296)
(187,295)(188,294)(189,293)(190,292)(191,291)(192,283)(193,289)(194,288)
(195,287)(196,286)(197,285)(198,284)(199,332)(200,338)(201,337)(202,336)
(203,335)(204,334)(205,333)(206,325)(207,331)(208,330)(209,329)(210,328)
(211,327)(212,326)(213,318)(214,324)(215,323)(216,322)(217,321)(218,320)
(219,319)(220,311)(221,317)(222,316)(223,315)(224,314)(225,313)(226,312);;
s2 := ( 3, 4)( 5, 9)( 6, 8)( 10, 11)( 12, 16)( 13, 15)( 17, 18)( 19, 23)
( 20, 22)( 24, 25)( 26, 30)( 27, 29)( 31, 39)( 32, 38)( 33, 44)( 34, 43)
( 35, 42)( 36, 41)( 37, 40)( 45, 53)( 46, 52)( 47, 58)( 48, 57)( 49, 56)
( 50, 55)( 51, 54)( 59, 74)( 60, 73)( 61, 79)( 62, 78)( 63, 77)( 64, 76)
( 65, 75)( 66, 81)( 67, 80)( 68, 86)( 69, 85)( 70, 84)( 71, 83)( 72, 82)
( 87,109)( 88,108)( 89,114)( 90,113)( 91,112)( 92,111)( 93,110)( 94,102)
( 95,101)( 96,107)( 97,106)( 98,105)( 99,104)(100,103)(115,144)(116,143)
(117,149)(118,148)(119,147)(120,146)(121,145)(122,151)(123,150)(124,156)
(125,155)(126,154)(127,153)(128,152)(129,158)(130,157)(131,163)(132,162)
(133,161)(134,160)(135,159)(136,165)(137,164)(138,170)(139,169)(140,168)
(141,167)(142,166)(171,214)(172,213)(173,219)(174,218)(175,217)(176,216)
(177,215)(178,221)(179,220)(180,226)(181,225)(182,224)(183,223)(184,222)
(185,200)(186,199)(187,205)(188,204)(189,203)(190,202)(191,201)(192,207)
(193,206)(194,212)(195,211)(196,210)(197,209)(198,208)(227,284)(228,283)
(229,289)(230,288)(231,287)(232,286)(233,285)(234,291)(235,290)(236,296)
(237,295)(238,294)(239,293)(240,292)(241,298)(242,297)(243,303)(244,302)
(245,301)(246,300)(247,299)(248,305)(249,304)(250,310)(251,309)(252,308)
(253,307)(254,306)(255,319)(256,318)(257,324)(258,323)(259,322)(260,321)
(261,320)(262,312)(263,311)(264,317)(265,316)(266,315)(267,314)(268,313)
(269,333)(270,332)(271,338)(272,337)(273,336)(274,335)(275,334)(276,326)
(277,325)(278,331)(279,330)(280,329)(281,328)(282,327)(339,431)(340,430)
(341,436)(342,435)(343,434)(344,433)(345,432)(346,424)(347,423)(348,429)
(349,428)(350,427)(351,426)(352,425)(353,445)(354,444)(355,450)(356,449)
(357,448)(358,447)(359,446)(360,438)(361,437)(362,443)(363,442)(364,441)
(365,440)(366,439)(367,403)(368,402)(369,408)(370,407)(371,406)(372,405)
(373,404)(374,396)(375,395)(376,401)(377,400)(378,399)(379,398)(380,397)
(381,417)(382,416)(383,422)(384,421)(385,420)(386,419)(387,418)(388,410)
(389,409)(390,415)(391,414)(392,413)(393,412)(394,411);;
s3 := ( 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)( 29,141)( 30,142)( 31,150)( 32,151)( 33,152)( 34,153)
( 35,154)( 36,155)( 37,156)( 38,143)( 39,144)( 40,145)( 41,146)( 42,147)
( 43,148)( 44,149)( 45,164)( 46,165)( 47,166)( 48,167)( 49,168)( 50,169)
( 51,170)( 52,157)( 53,158)( 54,159)( 55,160)( 56,161)( 57,162)( 58,163)
( 59,178)( 60,179)( 61,180)( 62,181)( 63,182)( 64,183)( 65,184)( 66,171)
( 67,172)( 68,173)( 69,174)( 70,175)( 71,176)( 72,177)( 73,192)( 74,193)
( 75,194)( 76,195)( 77,196)( 78,197)( 79,198)( 80,185)( 81,186)( 82,187)
( 83,188)( 84,189)( 85,190)( 86,191)( 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)(113,225)(114,226)
(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)(253,365)(254,366)(255,374)(256,375)(257,376)(258,377)
(259,378)(260,379)(261,380)(262,367)(263,368)(264,369)(265,370)(266,371)
(267,372)(268,373)(269,388)(270,389)(271,390)(272,391)(273,392)(274,393)
(275,394)(276,381)(277,382)(278,383)(279,384)(280,385)(281,386)(282,387)
(283,402)(284,403)(285,404)(286,405)(287,406)(288,407)(289,408)(290,395)
(291,396)(292,397)(293,398)(294,399)(295,400)(296,401)(297,416)(298,417)
(299,418)(300,419)(301,420)(302,421)(303,422)(304,409)(305,410)(306,411)
(307,412)(308,413)(309,414)(310,415)(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)(337,449)(338,450);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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(450)!(1,2);
s1 := Sym(450)!( 3,339)( 4,345)( 5,344)( 6,343)( 7,342)( 8,341)( 9,340)
( 10,346)( 11,352)( 12,351)( 13,350)( 14,349)( 15,348)( 16,347)( 17,353)
( 18,359)( 19,358)( 20,357)( 21,356)( 22,355)( 23,354)( 24,360)( 25,366)
( 26,365)( 27,364)( 28,363)( 29,362)( 30,361)( 31,367)( 32,373)( 33,372)
( 34,371)( 35,370)( 36,369)( 37,368)( 38,374)( 39,380)( 40,379)( 41,378)
( 42,377)( 43,376)( 44,375)( 45,381)( 46,387)( 47,386)( 48,385)( 49,384)
( 50,383)( 51,382)( 52,388)( 53,394)( 54,393)( 55,392)( 56,391)( 57,390)
( 58,389)( 59,416)( 60,422)( 61,421)( 62,420)( 63,419)( 64,418)( 65,417)
( 66,409)( 67,415)( 68,414)( 69,413)( 70,412)( 71,411)( 72,410)( 73,402)
( 74,408)( 75,407)( 76,406)( 77,405)( 78,404)( 79,403)( 80,395)( 81,401)
( 82,400)( 83,399)( 84,398)( 85,397)( 86,396)( 87,444)( 88,450)( 89,449)
( 90,448)( 91,447)( 92,446)( 93,445)( 94,437)( 95,443)( 96,442)( 97,441)
( 98,440)( 99,439)(100,438)(101,430)(102,436)(103,435)(104,434)(105,433)
(106,432)(107,431)(108,423)(109,429)(110,428)(111,427)(112,426)(113,425)
(114,424)(115,227)(116,233)(117,232)(118,231)(119,230)(120,229)(121,228)
(122,234)(123,240)(124,239)(125,238)(126,237)(127,236)(128,235)(129,241)
(130,247)(131,246)(132,245)(133,244)(134,243)(135,242)(136,248)(137,254)
(138,253)(139,252)(140,251)(141,250)(142,249)(143,255)(144,261)(145,260)
(146,259)(147,258)(148,257)(149,256)(150,262)(151,268)(152,267)(153,266)
(154,265)(155,264)(156,263)(157,269)(158,275)(159,274)(160,273)(161,272)
(162,271)(163,270)(164,276)(165,282)(166,281)(167,280)(168,279)(169,278)
(170,277)(171,304)(172,310)(173,309)(174,308)(175,307)(176,306)(177,305)
(178,297)(179,303)(180,302)(181,301)(182,300)(183,299)(184,298)(185,290)
(186,296)(187,295)(188,294)(189,293)(190,292)(191,291)(192,283)(193,289)
(194,288)(195,287)(196,286)(197,285)(198,284)(199,332)(200,338)(201,337)
(202,336)(203,335)(204,334)(205,333)(206,325)(207,331)(208,330)(209,329)
(210,328)(211,327)(212,326)(213,318)(214,324)(215,323)(216,322)(217,321)
(218,320)(219,319)(220,311)(221,317)(222,316)(223,315)(224,314)(225,313)
(226,312);
s2 := Sym(450)!( 3, 4)( 5, 9)( 6, 8)( 10, 11)( 12, 16)( 13, 15)( 17, 18)
( 19, 23)( 20, 22)( 24, 25)( 26, 30)( 27, 29)( 31, 39)( 32, 38)( 33, 44)
( 34, 43)( 35, 42)( 36, 41)( 37, 40)( 45, 53)( 46, 52)( 47, 58)( 48, 57)
( 49, 56)( 50, 55)( 51, 54)( 59, 74)( 60, 73)( 61, 79)( 62, 78)( 63, 77)
( 64, 76)( 65, 75)( 66, 81)( 67, 80)( 68, 86)( 69, 85)( 70, 84)( 71, 83)
( 72, 82)( 87,109)( 88,108)( 89,114)( 90,113)( 91,112)( 92,111)( 93,110)
( 94,102)( 95,101)( 96,107)( 97,106)( 98,105)( 99,104)(100,103)(115,144)
(116,143)(117,149)(118,148)(119,147)(120,146)(121,145)(122,151)(123,150)
(124,156)(125,155)(126,154)(127,153)(128,152)(129,158)(130,157)(131,163)
(132,162)(133,161)(134,160)(135,159)(136,165)(137,164)(138,170)(139,169)
(140,168)(141,167)(142,166)(171,214)(172,213)(173,219)(174,218)(175,217)
(176,216)(177,215)(178,221)(179,220)(180,226)(181,225)(182,224)(183,223)
(184,222)(185,200)(186,199)(187,205)(188,204)(189,203)(190,202)(191,201)
(192,207)(193,206)(194,212)(195,211)(196,210)(197,209)(198,208)(227,284)
(228,283)(229,289)(230,288)(231,287)(232,286)(233,285)(234,291)(235,290)
(236,296)(237,295)(238,294)(239,293)(240,292)(241,298)(242,297)(243,303)
(244,302)(245,301)(246,300)(247,299)(248,305)(249,304)(250,310)(251,309)
(252,308)(253,307)(254,306)(255,319)(256,318)(257,324)(258,323)(259,322)
(260,321)(261,320)(262,312)(263,311)(264,317)(265,316)(266,315)(267,314)
(268,313)(269,333)(270,332)(271,338)(272,337)(273,336)(274,335)(275,334)
(276,326)(277,325)(278,331)(279,330)(280,329)(281,328)(282,327)(339,431)
(340,430)(341,436)(342,435)(343,434)(344,433)(345,432)(346,424)(347,423)
(348,429)(349,428)(350,427)(351,426)(352,425)(353,445)(354,444)(355,450)
(356,449)(357,448)(358,447)(359,446)(360,438)(361,437)(362,443)(363,442)
(364,441)(365,440)(366,439)(367,403)(368,402)(369,408)(370,407)(371,406)
(372,405)(373,404)(374,396)(375,395)(376,401)(377,400)(378,399)(379,398)
(380,397)(381,417)(382,416)(383,422)(384,421)(385,420)(386,419)(387,418)
(388,410)(389,409)(390,415)(391,414)(392,413)(393,412)(394,411);
s3 := Sym(450)!( 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)( 29,141)( 30,142)( 31,150)( 32,151)( 33,152)
( 34,153)( 35,154)( 36,155)( 37,156)( 38,143)( 39,144)( 40,145)( 41,146)
( 42,147)( 43,148)( 44,149)( 45,164)( 46,165)( 47,166)( 48,167)( 49,168)
( 50,169)( 51,170)( 52,157)( 53,158)( 54,159)( 55,160)( 56,161)( 57,162)
( 58,163)( 59,178)( 60,179)( 61,180)( 62,181)( 63,182)( 64,183)( 65,184)
( 66,171)( 67,172)( 68,173)( 69,174)( 70,175)( 71,176)( 72,177)( 73,192)
( 74,193)( 75,194)( 76,195)( 77,196)( 78,197)( 79,198)( 80,185)( 81,186)
( 82,187)( 83,188)( 84,189)( 85,190)( 86,191)( 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)(113,225)
(114,226)(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)(253,365)(254,366)(255,374)(256,375)(257,376)
(258,377)(259,378)(260,379)(261,380)(262,367)(263,368)(264,369)(265,370)
(266,371)(267,372)(268,373)(269,388)(270,389)(271,390)(272,391)(273,392)
(274,393)(275,394)(276,381)(277,382)(278,383)(279,384)(280,385)(281,386)
(282,387)(283,402)(284,403)(285,404)(286,405)(287,406)(288,407)(289,408)
(290,395)(291,396)(292,397)(293,398)(294,399)(295,400)(296,401)(297,416)
(298,417)(299,418)(300,419)(301,420)(302,421)(303,422)(304,409)(305,410)
(306,411)(307,412)(308,413)(309,414)(310,415)(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)(337,449)
(338,450);
poly := sub<Sym(450)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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 >;
to this polytope