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# Polytope of Type {8,56}

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