Polytope of Type {2,56,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,56,8}*1792d
if this polytope has a name.
Group : SmallGroup(1792,150684)
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}*896b, {2,28,8}*896b
4-fold quotients : {2,28,4}*448
7-fold quotients : {2,8,8}*256d
8-fold quotients : {2,28,2}*224, {2,14,4}*224
14-fold quotients : {2,4,8}*128b, {2,8,4}*128b
16-fold quotients : {2,14,2}*112
28-fold quotients : {2,4,4}*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 := ( 3,227)( 4,233)( 5,232)( 6,231)( 7,230)( 8,229)( 9,228)( 10,234)
( 11,240)( 12,239)( 13,238)( 14,237)( 15,236)( 16,235)( 17,241)( 18,247)
( 19,246)( 20,245)( 21,244)( 22,243)( 23,242)( 24,248)( 25,254)( 26,253)
( 27,252)( 28,251)( 29,250)( 30,249)( 31,262)( 32,268)( 33,267)( 34,266)
( 35,265)( 36,264)( 37,263)( 38,255)( 39,261)( 40,260)( 41,259)( 42,258)
( 43,257)( 44,256)( 45,276)( 46,282)( 47,281)( 48,280)( 49,279)( 50,278)
( 51,277)( 52,269)( 53,275)( 54,274)( 55,273)( 56,272)( 57,271)( 58,270)
( 59,297)( 60,303)( 61,302)( 62,301)( 63,300)( 64,299)( 65,298)( 66,304)
( 67,310)( 68,309)( 69,308)( 70,307)( 71,306)( 72,305)( 73,283)( 74,289)
( 75,288)( 76,287)( 77,286)( 78,285)( 79,284)( 80,290)( 81,296)( 82,295)
( 83,294)( 84,293)( 85,292)( 86,291)( 87,332)( 88,338)( 89,337)( 90,336)
( 91,335)( 92,334)( 93,333)( 94,325)( 95,331)( 96,330)( 97,329)( 98,328)
( 99,327)(100,326)(101,318)(102,324)(103,323)(104,322)(105,321)(106,320)
(107,319)(108,311)(109,317)(110,316)(111,315)(112,314)(113,313)(114,312)
(115,339)(116,345)(117,344)(118,343)(119,342)(120,341)(121,340)(122,346)
(123,352)(124,351)(125,350)(126,349)(127,348)(128,347)(129,353)(130,359)
(131,358)(132,357)(133,356)(134,355)(135,354)(136,360)(137,366)(138,365)
(139,364)(140,363)(141,362)(142,361)(143,374)(144,380)(145,379)(146,378)
(147,377)(148,376)(149,375)(150,367)(151,373)(152,372)(153,371)(154,370)
(155,369)(156,368)(157,388)(158,394)(159,393)(160,392)(161,391)(162,390)
(163,389)(164,381)(165,387)(166,386)(167,385)(168,384)(169,383)(170,382)
(171,409)(172,415)(173,414)(174,413)(175,412)(176,411)(177,410)(178,416)
(179,422)(180,421)(181,420)(182,419)(183,418)(184,417)(185,395)(186,401)
(187,400)(188,399)(189,398)(190,397)(191,396)(192,402)(193,408)(194,407)
(195,406)(196,405)(197,404)(198,403)(199,444)(200,450)(201,449)(202,448)
(203,447)(204,446)(205,445)(206,437)(207,443)(208,442)(209,441)(210,440)
(211,439)(212,438)(213,430)(214,436)(215,435)(216,434)(217,433)(218,432)
(219,431)(220,423)(221,429)(222,428)(223,427)(224,426)(225,425)(226,424);;
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, 53)( 32, 52)( 33, 58)( 34, 57)
( 35, 56)( 36, 55)( 37, 54)( 38, 46)( 39, 45)( 40, 51)( 41, 50)( 42, 49)
( 43, 48)( 44, 47)( 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, 95)( 88, 94)( 89,100)( 90, 99)( 91, 98)( 92, 97)( 93, 96)(101,109)
(102,108)(103,114)(104,113)(105,112)(106,111)(107,110)(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)(255,333)(256,332)(257,338)(258,337)(259,336)(260,335)
(261,334)(262,326)(263,325)(264,331)(265,330)(266,329)(267,328)(268,327)
(269,319)(270,318)(271,324)(272,323)(273,322)(274,321)(275,320)(276,312)
(277,311)(278,317)(279,316)(280,315)(281,314)(282,313)(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)( 31,164)( 32,165)( 33,166)( 34,167)
( 35,168)( 36,169)( 37,170)( 38,157)( 39,158)( 40,159)( 41,160)( 42,161)
( 43,162)( 44,163)( 45,150)( 46,151)( 47,152)( 48,153)( 49,154)( 50,155)
( 51,156)( 52,143)( 53,144)( 54,145)( 55,146)( 56,147)( 57,148)( 58,149)
( 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)( 87,213)( 88,214)( 89,215)( 90,216)
( 91,217)( 92,218)( 93,219)( 94,220)( 95,221)( 96,222)( 97,223)( 98,224)
( 99,225)(100,226)(101,199)(102,200)(103,201)(104,202)(105,203)(106,204)
(107,205)(108,206)(109,207)(110,208)(111,209)(112,210)(113,211)(114,212)
(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)(255,388)(256,389)(257,390)(258,391)
(259,392)(260,393)(261,394)(262,381)(263,382)(264,383)(265,384)(266,385)
(267,386)(268,387)(269,374)(270,375)(271,376)(272,377)(273,378)(274,379)
(275,380)(276,367)(277,368)(278,369)(279,370)(280,371)(281,372)(282,373)
(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)(311,437)(312,438)(313,439)(314,440)
(315,441)(316,442)(317,443)(318,444)(319,445)(320,446)(321,447)(322,448)
(323,449)(324,450)(325,423)(326,424)(327,425)(328,426)(329,427)(330,428)
(331,429)(332,430)(333,431)(334,432)(335,433)(336,434)(337,435)(338,436);;
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,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*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*s3*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s2*s1*s2*s3*s1*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(450)!(1,2);
s1 := Sym(450)!( 3,227)( 4,233)( 5,232)( 6,231)( 7,230)( 8,229)( 9,228)
( 10,234)( 11,240)( 12,239)( 13,238)( 14,237)( 15,236)( 16,235)( 17,241)
( 18,247)( 19,246)( 20,245)( 21,244)( 22,243)( 23,242)( 24,248)( 25,254)
( 26,253)( 27,252)( 28,251)( 29,250)( 30,249)( 31,262)( 32,268)( 33,267)
( 34,266)( 35,265)( 36,264)( 37,263)( 38,255)( 39,261)( 40,260)( 41,259)
( 42,258)( 43,257)( 44,256)( 45,276)( 46,282)( 47,281)( 48,280)( 49,279)
( 50,278)( 51,277)( 52,269)( 53,275)( 54,274)( 55,273)( 56,272)( 57,271)
( 58,270)( 59,297)( 60,303)( 61,302)( 62,301)( 63,300)( 64,299)( 65,298)
( 66,304)( 67,310)( 68,309)( 69,308)( 70,307)( 71,306)( 72,305)( 73,283)
( 74,289)( 75,288)( 76,287)( 77,286)( 78,285)( 79,284)( 80,290)( 81,296)
( 82,295)( 83,294)( 84,293)( 85,292)( 86,291)( 87,332)( 88,338)( 89,337)
( 90,336)( 91,335)( 92,334)( 93,333)( 94,325)( 95,331)( 96,330)( 97,329)
( 98,328)( 99,327)(100,326)(101,318)(102,324)(103,323)(104,322)(105,321)
(106,320)(107,319)(108,311)(109,317)(110,316)(111,315)(112,314)(113,313)
(114,312)(115,339)(116,345)(117,344)(118,343)(119,342)(120,341)(121,340)
(122,346)(123,352)(124,351)(125,350)(126,349)(127,348)(128,347)(129,353)
(130,359)(131,358)(132,357)(133,356)(134,355)(135,354)(136,360)(137,366)
(138,365)(139,364)(140,363)(141,362)(142,361)(143,374)(144,380)(145,379)
(146,378)(147,377)(148,376)(149,375)(150,367)(151,373)(152,372)(153,371)
(154,370)(155,369)(156,368)(157,388)(158,394)(159,393)(160,392)(161,391)
(162,390)(163,389)(164,381)(165,387)(166,386)(167,385)(168,384)(169,383)
(170,382)(171,409)(172,415)(173,414)(174,413)(175,412)(176,411)(177,410)
(178,416)(179,422)(180,421)(181,420)(182,419)(183,418)(184,417)(185,395)
(186,401)(187,400)(188,399)(189,398)(190,397)(191,396)(192,402)(193,408)
(194,407)(195,406)(196,405)(197,404)(198,403)(199,444)(200,450)(201,449)
(202,448)(203,447)(204,446)(205,445)(206,437)(207,443)(208,442)(209,441)
(210,440)(211,439)(212,438)(213,430)(214,436)(215,435)(216,434)(217,433)
(218,432)(219,431)(220,423)(221,429)(222,428)(223,427)(224,426)(225,425)
(226,424);
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, 53)( 32, 52)( 33, 58)
( 34, 57)( 35, 56)( 36, 55)( 37, 54)( 38, 46)( 39, 45)( 40, 51)( 41, 50)
( 42, 49)( 43, 48)( 44, 47)( 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, 95)( 88, 94)( 89,100)( 90, 99)( 91, 98)( 92, 97)( 93, 96)
(101,109)(102,108)(103,114)(104,113)(105,112)(106,111)(107,110)(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)(255,333)(256,332)(257,338)(258,337)(259,336)
(260,335)(261,334)(262,326)(263,325)(264,331)(265,330)(266,329)(267,328)
(268,327)(269,319)(270,318)(271,324)(272,323)(273,322)(274,321)(275,320)
(276,312)(277,311)(278,317)(279,316)(280,315)(281,314)(282,313)(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)( 31,164)( 32,165)( 33,166)
( 34,167)( 35,168)( 36,169)( 37,170)( 38,157)( 39,158)( 40,159)( 41,160)
( 42,161)( 43,162)( 44,163)( 45,150)( 46,151)( 47,152)( 48,153)( 49,154)
( 50,155)( 51,156)( 52,143)( 53,144)( 54,145)( 55,146)( 56,147)( 57,148)
( 58,149)( 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)( 87,213)( 88,214)( 89,215)
( 90,216)( 91,217)( 92,218)( 93,219)( 94,220)( 95,221)( 96,222)( 97,223)
( 98,224)( 99,225)(100,226)(101,199)(102,200)(103,201)(104,202)(105,203)
(106,204)(107,205)(108,206)(109,207)(110,208)(111,209)(112,210)(113,211)
(114,212)(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)(255,388)(256,389)(257,390)
(258,391)(259,392)(260,393)(261,394)(262,381)(263,382)(264,383)(265,384)
(266,385)(267,386)(268,387)(269,374)(270,375)(271,376)(272,377)(273,378)
(274,379)(275,380)(276,367)(277,368)(278,369)(279,370)(280,371)(281,372)
(282,373)(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)(311,437)(312,438)(313,439)
(314,440)(315,441)(316,442)(317,443)(318,444)(319,445)(320,446)(321,447)
(322,448)(323,449)(324,450)(325,423)(326,424)(327,425)(328,426)(329,427)
(330,428)(331,429)(332,430)(333,431)(334,432)(335,433)(336,434)(337,435)
(338,436);
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, s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*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*s3*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s2*s1*s2*s3*s1*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 >;
to this polytope