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# Polytope of Type {56,8}

Atlas Canonical Name : {56,8}*896d
if this polytope has a name.
Group : SmallGroup(896,910)
Rank : 3
Schlafli Type : {56,8}
Number of vertices, edges, etc : 56, 224, 8
Order of s0s1s2 : 56
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
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}*448b, {28,8}*448a
4-fold quotients : {28,4}*224, {14,8}*224
7-fold quotients : {8,8}*128a
8-fold quotients : {28,2}*112, {14,4}*112
14-fold quotients : {4,8}*64a, {8,4}*64b
16-fold quotients : {14,2}*56
28-fold quotients : {4,4}*32, {2,8}*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}*1792a, {112,8}*1792b, {56,16}*1792c, {56,16}*1792e
Permutation Representation (GAP) :
```s0 := (  1,113)(  2,119)(  3,118)(  4,117)(  5,116)(  6,115)(  7,114)(  8,120)
(  9,126)( 10,125)( 11,124)( 12,123)( 13,122)( 14,121)( 15,127)( 16,133)
( 17,132)( 18,131)( 19,130)( 20,129)( 21,128)( 22,134)( 23,140)( 24,139)
( 25,138)( 26,137)( 27,136)( 28,135)( 29,148)( 30,154)( 31,153)( 32,152)
( 33,151)( 34,150)( 35,149)( 36,141)( 37,147)( 38,146)( 39,145)( 40,144)
( 41,143)( 42,142)( 43,162)( 44,168)( 45,167)( 46,166)( 47,165)( 48,164)
( 49,163)( 50,155)( 51,161)( 52,160)( 53,159)( 54,158)( 55,157)( 56,156)
( 57,176)( 58,182)( 59,181)( 60,180)( 61,179)( 62,178)( 63,177)( 64,169)
( 65,175)( 66,174)( 67,173)( 68,172)( 69,171)( 70,170)( 71,190)( 72,196)
( 73,195)( 74,194)( 75,193)( 76,192)( 77,191)( 78,183)( 79,189)( 80,188)
( 81,187)( 82,186)( 83,185)( 84,184)( 85,197)( 86,203)( 87,202)( 88,201)
( 89,200)( 90,199)( 91,198)( 92,204)( 93,210)( 94,209)( 95,208)( 96,207)
( 97,206)( 98,205)( 99,211)(100,217)(101,216)(102,215)(103,214)(104,213)
(105,212)(106,218)(107,224)(108,223)(109,222)(110,221)(111,220)(112,219)
(225,337)(226,343)(227,342)(228,341)(229,340)(230,339)(231,338)(232,344)
(233,350)(234,349)(235,348)(236,347)(237,346)(238,345)(239,351)(240,357)
(241,356)(242,355)(243,354)(244,353)(245,352)(246,358)(247,364)(248,363)
(249,362)(250,361)(251,360)(252,359)(253,372)(254,378)(255,377)(256,376)
(257,375)(258,374)(259,373)(260,365)(261,371)(262,370)(263,369)(264,368)
(265,367)(266,366)(267,386)(268,392)(269,391)(270,390)(271,389)(272,388)
(273,387)(274,379)(275,385)(276,384)(277,383)(278,382)(279,381)(280,380)
(281,400)(282,406)(283,405)(284,404)(285,403)(286,402)(287,401)(288,393)
(289,399)(290,398)(291,397)(292,396)(293,395)(294,394)(295,414)(296,420)
(297,419)(298,418)(299,417)(300,416)(301,415)(302,407)(303,413)(304,412)
(305,411)(306,410)(307,409)(308,408)(309,421)(310,427)(311,426)(312,425)
(313,424)(314,423)(315,422)(316,428)(317,434)(318,433)(319,432)(320,431)
(321,430)(322,429)(323,435)(324,441)(325,440)(326,439)(327,438)(328,437)
(329,436)(330,442)(331,448)(332,447)(333,446)(334,445)(335,444)(336,443);;
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 := (  1,225)(  2,226)(  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)( 29,260)( 30,261)( 31,262)( 32,263)
( 33,264)( 34,265)( 35,266)( 36,253)( 37,254)( 38,255)( 39,256)( 40,257)
( 41,258)( 42,259)( 43,274)( 44,275)( 45,276)( 46,277)( 47,278)( 48,279)
( 49,280)( 50,267)( 51,268)( 52,269)( 53,270)( 54,271)( 55,272)( 56,273)
( 57,295)( 58,296)( 59,297)( 60,298)( 61,299)( 62,300)( 63,301)( 64,302)
( 65,303)( 66,304)( 67,305)( 68,306)( 69,307)( 70,308)( 71,281)( 72,282)
( 73,283)( 74,284)( 75,285)( 76,286)( 77,287)( 78,288)( 79,289)( 80,290)
( 81,291)( 82,292)( 83,293)( 84,294)( 85,330)( 86,331)( 87,332)( 88,333)
( 89,334)( 90,335)( 91,336)( 92,323)( 93,324)( 94,325)( 95,326)( 96,327)
( 97,328)( 98,329)( 99,316)(100,317)(101,318)(102,319)(103,320)(104,321)
(105,322)(106,309)(107,310)(108,311)(109,312)(110,313)(111,314)(112,315)
(113,337)(114,338)(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)(141,372)(142,373)(143,374)(144,375)
(145,376)(146,377)(147,378)(148,365)(149,366)(150,367)(151,368)(152,369)
(153,370)(154,371)(155,386)(156,387)(157,388)(158,389)(159,390)(160,391)
(161,392)(162,379)(163,380)(164,381)(165,382)(166,383)(167,384)(168,385)
(169,407)(170,408)(171,409)(172,410)(173,411)(174,412)(175,413)(176,414)
(177,415)(178,416)(179,417)(180,418)(181,419)(182,420)(183,393)(184,394)
(185,395)(186,396)(187,397)(188,398)(189,399)(190,400)(191,401)(192,402)
(193,403)(194,404)(195,405)(196,406)(197,442)(198,443)(199,444)(200,445)
(201,446)(202,447)(203,448)(204,435)(205,436)(206,437)(207,438)(208,439)
(209,440)(210,441)(211,428)(212,429)(213,430)(214,431)(215,432)(216,433)
(217,434)(218,421)(219,422)(220,423)(221,424)(222,425)(223,426)(224,427);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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*s2*s1*s0*s1*s0*s2*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)!(  1,113)(  2,119)(  3,118)(  4,117)(  5,116)(  6,115)(  7,114)
(  8,120)(  9,126)( 10,125)( 11,124)( 12,123)( 13,122)( 14,121)( 15,127)
( 16,133)( 17,132)( 18,131)( 19,130)( 20,129)( 21,128)( 22,134)( 23,140)
( 24,139)( 25,138)( 26,137)( 27,136)( 28,135)( 29,148)( 30,154)( 31,153)
( 32,152)( 33,151)( 34,150)( 35,149)( 36,141)( 37,147)( 38,146)( 39,145)
( 40,144)( 41,143)( 42,142)( 43,162)( 44,168)( 45,167)( 46,166)( 47,165)
( 48,164)( 49,163)( 50,155)( 51,161)( 52,160)( 53,159)( 54,158)( 55,157)
( 56,156)( 57,176)( 58,182)( 59,181)( 60,180)( 61,179)( 62,178)( 63,177)
( 64,169)( 65,175)( 66,174)( 67,173)( 68,172)( 69,171)( 70,170)( 71,190)
( 72,196)( 73,195)( 74,194)( 75,193)( 76,192)( 77,191)( 78,183)( 79,189)
( 80,188)( 81,187)( 82,186)( 83,185)( 84,184)( 85,197)( 86,203)( 87,202)
( 88,201)( 89,200)( 90,199)( 91,198)( 92,204)( 93,210)( 94,209)( 95,208)
( 96,207)( 97,206)( 98,205)( 99,211)(100,217)(101,216)(102,215)(103,214)
(104,213)(105,212)(106,218)(107,224)(108,223)(109,222)(110,221)(111,220)
(112,219)(225,337)(226,343)(227,342)(228,341)(229,340)(230,339)(231,338)
(232,344)(233,350)(234,349)(235,348)(236,347)(237,346)(238,345)(239,351)
(240,357)(241,356)(242,355)(243,354)(244,353)(245,352)(246,358)(247,364)
(248,363)(249,362)(250,361)(251,360)(252,359)(253,372)(254,378)(255,377)
(256,376)(257,375)(258,374)(259,373)(260,365)(261,371)(262,370)(263,369)
(264,368)(265,367)(266,366)(267,386)(268,392)(269,391)(270,390)(271,389)
(272,388)(273,387)(274,379)(275,385)(276,384)(277,383)(278,382)(279,381)
(280,380)(281,400)(282,406)(283,405)(284,404)(285,403)(286,402)(287,401)
(288,393)(289,399)(290,398)(291,397)(292,396)(293,395)(294,394)(295,414)
(296,420)(297,419)(298,418)(299,417)(300,416)(301,415)(302,407)(303,413)
(304,412)(305,411)(306,410)(307,409)(308,408)(309,421)(310,427)(311,426)
(312,425)(313,424)(314,423)(315,422)(316,428)(317,434)(318,433)(319,432)
(320,431)(321,430)(322,429)(323,435)(324,441)(325,440)(326,439)(327,438)
(328,437)(329,436)(330,442)(331,448)(332,447)(333,446)(334,445)(335,444)
(336,443);
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)!(  1,225)(  2,226)(  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)( 29,260)( 30,261)( 31,262)
( 32,263)( 33,264)( 34,265)( 35,266)( 36,253)( 37,254)( 38,255)( 39,256)
( 40,257)( 41,258)( 42,259)( 43,274)( 44,275)( 45,276)( 46,277)( 47,278)
( 48,279)( 49,280)( 50,267)( 51,268)( 52,269)( 53,270)( 54,271)( 55,272)
( 56,273)( 57,295)( 58,296)( 59,297)( 60,298)( 61,299)( 62,300)( 63,301)
( 64,302)( 65,303)( 66,304)( 67,305)( 68,306)( 69,307)( 70,308)( 71,281)
( 72,282)( 73,283)( 74,284)( 75,285)( 76,286)( 77,287)( 78,288)( 79,289)
( 80,290)( 81,291)( 82,292)( 83,293)( 84,294)( 85,330)( 86,331)( 87,332)
( 88,333)( 89,334)( 90,335)( 91,336)( 92,323)( 93,324)( 94,325)( 95,326)
( 96,327)( 97,328)( 98,329)( 99,316)(100,317)(101,318)(102,319)(103,320)
(104,321)(105,322)(106,309)(107,310)(108,311)(109,312)(110,313)(111,314)
(112,315)(113,337)(114,338)(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)(141,372)(142,373)(143,374)
(144,375)(145,376)(146,377)(147,378)(148,365)(149,366)(150,367)(151,368)
(152,369)(153,370)(154,371)(155,386)(156,387)(157,388)(158,389)(159,390)
(160,391)(161,392)(162,379)(163,380)(164,381)(165,382)(166,383)(167,384)
(168,385)(169,407)(170,408)(171,409)(172,410)(173,411)(174,412)(175,413)
(176,414)(177,415)(178,416)(179,417)(180,418)(181,419)(182,420)(183,393)
(184,394)(185,395)(186,396)(187,397)(188,398)(189,399)(190,400)(191,401)
(192,402)(193,403)(194,404)(195,405)(196,406)(197,442)(198,443)(199,444)
(200,445)(201,446)(202,447)(203,448)(204,435)(205,436)(206,437)(207,438)
(208,439)(209,440)(210,441)(211,428)(212,429)(213,430)(214,431)(215,432)
(216,433)(217,434)(218,421)(219,422)(220,423)(221,424)(222,425)(223,426)
(224,427);
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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*s2*s1*s0*s1*s0*s2*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