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# Polytope of Type {2,56,8}

Atlas Canonical Name : {2,56,8}*1792a
if this polytope has a name.
Group : SmallGroup(1792,145173)
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}*896a
4-fold quotients : {2,28,4}*448, {2,14,8}*448
7-fold quotients : {2,8,8}*256a
8-fold quotients : {2,28,2}*224, {2,14,4}*224
14-fold quotients : {2,4,8}*128a, {2,8,4}*128b
16-fold quotients : {2,14,2}*112
28-fold quotients : {2,4,4}*64, {2,2,8}*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 := (  3,115)(  4,121)(  5,120)(  6,119)(  7,118)(  8,117)(  9,116)( 10,122)
( 11,128)( 12,127)( 13,126)( 14,125)( 15,124)( 16,123)( 17,129)( 18,135)
( 19,134)( 20,133)( 21,132)( 22,131)( 23,130)( 24,136)( 25,142)( 26,141)
( 27,140)( 28,139)( 29,138)( 30,137)( 31,150)( 32,156)( 33,155)( 34,154)
( 35,153)( 36,152)( 37,151)( 38,143)( 39,149)( 40,148)( 41,147)( 42,146)
( 43,145)( 44,144)( 45,164)( 46,170)( 47,169)( 48,168)( 49,167)( 50,166)
( 51,165)( 52,157)( 53,163)( 54,162)( 55,161)( 56,160)( 57,159)( 58,158)
( 59,178)( 60,184)( 61,183)( 62,182)( 63,181)( 64,180)( 65,179)( 66,171)
( 67,177)( 68,176)( 69,175)( 70,174)( 71,173)( 72,172)( 73,192)( 74,198)
( 75,197)( 76,196)( 77,195)( 78,194)( 79,193)( 80,185)( 81,191)( 82,190)
( 83,189)( 84,188)( 85,187)( 86,186)( 87,199)( 88,205)( 89,204)( 90,203)
( 91,202)( 92,201)( 93,200)( 94,206)( 95,212)( 96,211)( 97,210)( 98,209)
( 99,208)(100,207)(101,213)(102,219)(103,218)(104,217)(105,216)(106,215)
(107,214)(108,220)(109,226)(110,225)(111,224)(112,223)(113,222)(114,221)
(227,339)(228,345)(229,344)(230,343)(231,342)(232,341)(233,340)(234,346)
(235,352)(236,351)(237,350)(238,349)(239,348)(240,347)(241,353)(242,359)
(243,358)(244,357)(245,356)(246,355)(247,354)(248,360)(249,366)(250,365)
(251,364)(252,363)(253,362)(254,361)(255,374)(256,380)(257,379)(258,378)
(259,377)(260,376)(261,375)(262,367)(263,373)(264,372)(265,371)(266,370)
(267,369)(268,368)(269,388)(270,394)(271,393)(272,392)(273,391)(274,390)
(275,389)(276,381)(277,387)(278,386)(279,385)(280,384)(281,383)(282,382)
(283,402)(284,408)(285,407)(286,406)(287,405)(288,404)(289,403)(290,395)
(291,401)(292,400)(293,399)(294,398)(295,397)(296,396)(297,416)(298,422)
(299,421)(300,420)(301,419)(302,418)(303,417)(304,409)(305,415)(306,414)
(307,413)(308,412)(309,411)(310,410)(311,423)(312,429)(313,428)(314,427)
(315,426)(316,425)(317,424)(318,430)(319,436)(320,435)(321,434)(322,433)
(323,432)(324,431)(325,437)(326,443)(327,442)(328,441)(329,440)(330,439)
(331,438)(332,444)(333,450)(334,449)(335,448)(336,447)(337,446)(338,445);;
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 := (  3,227)(  4,228)(  5,229)(  6,230)(  7,231)(  8,232)(  9,233)( 10,234)
( 11,235)( 12,236)( 13,237)( 14,238)( 15,239)( 16,240)( 17,241)( 18,242)
( 19,243)( 20,244)( 21,245)( 22,246)( 23,247)( 24,248)( 25,249)( 26,250)
( 27,251)( 28,252)( 29,253)( 30,254)( 31,262)( 32,263)( 33,264)( 34,265)
( 35,266)( 36,267)( 37,268)( 38,255)( 39,256)( 40,257)( 41,258)( 42,259)
( 43,260)( 44,261)( 45,276)( 46,277)( 47,278)( 48,279)( 49,280)( 50,281)
( 51,282)( 52,269)( 53,270)( 54,271)( 55,272)( 56,273)( 57,274)( 58,275)
( 59,297)( 60,298)( 61,299)( 62,300)( 63,301)( 64,302)( 65,303)( 66,304)
( 67,305)( 68,306)( 69,307)( 70,308)( 71,309)( 72,310)( 73,283)( 74,284)
( 75,285)( 76,286)( 77,287)( 78,288)( 79,289)( 80,290)( 81,291)( 82,292)
( 83,293)( 84,294)( 85,295)( 86,296)( 87,332)( 88,333)( 89,334)( 90,335)
( 91,336)( 92,337)( 93,338)( 94,325)( 95,326)( 96,327)( 97,328)( 98,329)
( 99,330)(100,331)(101,318)(102,319)(103,320)(104,321)(105,322)(106,323)
(107,324)(108,311)(109,312)(110,313)(111,314)(112,315)(113,316)(114,317)
(115,339)(116,340)(117,341)(118,342)(119,343)(120,344)(121,345)(122,346)
(123,347)(124,348)(125,349)(126,350)(127,351)(128,352)(129,353)(130,354)
(131,355)(132,356)(133,357)(134,358)(135,359)(136,360)(137,361)(138,362)
(139,363)(140,364)(141,365)(142,366)(143,374)(144,375)(145,376)(146,377)
(147,378)(148,379)(149,380)(150,367)(151,368)(152,369)(153,370)(154,371)
(155,372)(156,373)(157,388)(158,389)(159,390)(160,391)(161,392)(162,393)
(163,394)(164,381)(165,382)(166,383)(167,384)(168,385)(169,386)(170,387)
(171,409)(172,410)(173,411)(174,412)(175,413)(176,414)(177,415)(178,416)
(179,417)(180,418)(181,419)(182,420)(183,421)(184,422)(185,395)(186,396)
(187,397)(188,398)(189,399)(190,400)(191,401)(192,402)(193,403)(194,404)
(195,405)(196,406)(197,407)(198,408)(199,444)(200,445)(201,446)(202,447)
(203,448)(204,449)(205,450)(206,437)(207,438)(208,439)(209,440)(210,441)
(211,442)(212,443)(213,430)(214,431)(215,432)(216,433)(217,434)(218,435)
(219,436)(220,423)(221,424)(222,425)(223,426)(224,427)(225,428)(226,429);;
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*s3*s2*s1*s2*s1*s3*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)!(  3,115)(  4,121)(  5,120)(  6,119)(  7,118)(  8,117)(  9,116)
( 10,122)( 11,128)( 12,127)( 13,126)( 14,125)( 15,124)( 16,123)( 17,129)
( 18,135)( 19,134)( 20,133)( 21,132)( 22,131)( 23,130)( 24,136)( 25,142)
( 26,141)( 27,140)( 28,139)( 29,138)( 30,137)( 31,150)( 32,156)( 33,155)
( 34,154)( 35,153)( 36,152)( 37,151)( 38,143)( 39,149)( 40,148)( 41,147)
( 42,146)( 43,145)( 44,144)( 45,164)( 46,170)( 47,169)( 48,168)( 49,167)
( 50,166)( 51,165)( 52,157)( 53,163)( 54,162)( 55,161)( 56,160)( 57,159)
( 58,158)( 59,178)( 60,184)( 61,183)( 62,182)( 63,181)( 64,180)( 65,179)
( 66,171)( 67,177)( 68,176)( 69,175)( 70,174)( 71,173)( 72,172)( 73,192)
( 74,198)( 75,197)( 76,196)( 77,195)( 78,194)( 79,193)( 80,185)( 81,191)
( 82,190)( 83,189)( 84,188)( 85,187)( 86,186)( 87,199)( 88,205)( 89,204)
( 90,203)( 91,202)( 92,201)( 93,200)( 94,206)( 95,212)( 96,211)( 97,210)
( 98,209)( 99,208)(100,207)(101,213)(102,219)(103,218)(104,217)(105,216)
(106,215)(107,214)(108,220)(109,226)(110,225)(111,224)(112,223)(113,222)
(114,221)(227,339)(228,345)(229,344)(230,343)(231,342)(232,341)(233,340)
(234,346)(235,352)(236,351)(237,350)(238,349)(239,348)(240,347)(241,353)
(242,359)(243,358)(244,357)(245,356)(246,355)(247,354)(248,360)(249,366)
(250,365)(251,364)(252,363)(253,362)(254,361)(255,374)(256,380)(257,379)
(258,378)(259,377)(260,376)(261,375)(262,367)(263,373)(264,372)(265,371)
(266,370)(267,369)(268,368)(269,388)(270,394)(271,393)(272,392)(273,391)
(274,390)(275,389)(276,381)(277,387)(278,386)(279,385)(280,384)(281,383)
(282,382)(283,402)(284,408)(285,407)(286,406)(287,405)(288,404)(289,403)
(290,395)(291,401)(292,400)(293,399)(294,398)(295,397)(296,396)(297,416)
(298,422)(299,421)(300,420)(301,419)(302,418)(303,417)(304,409)(305,415)
(306,414)(307,413)(308,412)(309,411)(310,410)(311,423)(312,429)(313,428)
(314,427)(315,426)(316,425)(317,424)(318,430)(319,436)(320,435)(321,434)
(322,433)(323,432)(324,431)(325,437)(326,443)(327,442)(328,441)(329,440)
(330,439)(331,438)(332,444)(333,450)(334,449)(335,448)(336,447)(337,446)
(338,445);
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)!(  3,227)(  4,228)(  5,229)(  6,230)(  7,231)(  8,232)(  9,233)
( 10,234)( 11,235)( 12,236)( 13,237)( 14,238)( 15,239)( 16,240)( 17,241)
( 18,242)( 19,243)( 20,244)( 21,245)( 22,246)( 23,247)( 24,248)( 25,249)
( 26,250)( 27,251)( 28,252)( 29,253)( 30,254)( 31,262)( 32,263)( 33,264)
( 34,265)( 35,266)( 36,267)( 37,268)( 38,255)( 39,256)( 40,257)( 41,258)
( 42,259)( 43,260)( 44,261)( 45,276)( 46,277)( 47,278)( 48,279)( 49,280)
( 50,281)( 51,282)( 52,269)( 53,270)( 54,271)( 55,272)( 56,273)( 57,274)
( 58,275)( 59,297)( 60,298)( 61,299)( 62,300)( 63,301)( 64,302)( 65,303)
( 66,304)( 67,305)( 68,306)( 69,307)( 70,308)( 71,309)( 72,310)( 73,283)
( 74,284)( 75,285)( 76,286)( 77,287)( 78,288)( 79,289)( 80,290)( 81,291)
( 82,292)( 83,293)( 84,294)( 85,295)( 86,296)( 87,332)( 88,333)( 89,334)
( 90,335)( 91,336)( 92,337)( 93,338)( 94,325)( 95,326)( 96,327)( 97,328)
( 98,329)( 99,330)(100,331)(101,318)(102,319)(103,320)(104,321)(105,322)
(106,323)(107,324)(108,311)(109,312)(110,313)(111,314)(112,315)(113,316)
(114,317)(115,339)(116,340)(117,341)(118,342)(119,343)(120,344)(121,345)
(122,346)(123,347)(124,348)(125,349)(126,350)(127,351)(128,352)(129,353)
(130,354)(131,355)(132,356)(133,357)(134,358)(135,359)(136,360)(137,361)
(138,362)(139,363)(140,364)(141,365)(142,366)(143,374)(144,375)(145,376)
(146,377)(147,378)(148,379)(149,380)(150,367)(151,368)(152,369)(153,370)
(154,371)(155,372)(156,373)(157,388)(158,389)(159,390)(160,391)(161,392)
(162,393)(163,394)(164,381)(165,382)(166,383)(167,384)(168,385)(169,386)
(170,387)(171,409)(172,410)(173,411)(174,412)(175,413)(176,414)(177,415)
(178,416)(179,417)(180,418)(181,419)(182,420)(183,421)(184,422)(185,395)
(186,396)(187,397)(188,398)(189,399)(190,400)(191,401)(192,402)(193,403)
(194,404)(195,405)(196,406)(197,407)(198,408)(199,444)(200,445)(201,446)
(202,447)(203,448)(204,449)(205,450)(206,437)(207,438)(208,439)(209,440)
(210,441)(211,442)(212,443)(213,430)(214,431)(215,432)(216,433)(217,434)
(218,435)(219,436)(220,423)(221,424)(222,425)(223,426)(224,427)(225,428)
(226,429);
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*s3*s2*s1*s2*s1*s3*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