Polytope of Type {4,112}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,112}*896b
if this polytope has a name.
Group : SmallGroup(896,1640)
Rank : 3
Schlafli Type : {4,112}
Number of vertices, edges, etc : 4, 224, 112
Order of s0s1s2 : 112
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,112,2} of size 1792
Vertex Figure Of :
{2,4,112} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,56}*448a
4-fold quotients : {4,28}*224, {2,56}*224
7-fold quotients : {4,16}*128b
8-fold quotients : {2,28}*112, {4,14}*112
14-fold quotients : {4,8}*64a
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 : {4,112}*1792a, {8,112}*1792e, {8,112}*1792f
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)( 29,141)( 30,142)( 31,143)( 32,144)
( 33,145)( 34,146)( 35,147)( 36,148)( 37,149)( 38,150)( 39,151)( 40,152)
( 41,153)( 42,154)( 43,155)( 44,156)( 45,157)( 46,158)( 47,159)( 48,160)
( 49,161)( 50,162)( 51,163)( 52,164)( 53,165)( 54,166)( 55,167)( 56,168)
( 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)( 85,204)( 86,205)( 87,206)( 88,207)
( 89,208)( 90,209)( 91,210)( 92,197)( 93,198)( 94,199)( 95,200)( 96,201)
( 97,202)( 98,203)( 99,218)(100,219)(101,220)(102,221)(103,222)(104,223)
(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)
(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)(253,365)(254,366)(255,367)(256,368)
(257,369)(258,370)(259,371)(260,372)(261,373)(262,374)(263,375)(264,376)
(265,377)(266,378)(267,379)(268,380)(269,381)(270,382)(271,383)(272,384)
(273,385)(274,386)(275,387)(276,388)(277,389)(278,390)(279,391)(280,392)
(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)(309,428)(310,429)(311,430)(312,431)
(313,432)(314,433)(315,434)(316,421)(317,422)(318,423)(319,424)(320,425)
(321,426)(322,427)(323,442)(324,443)(325,444)(326,445)(327,446)(328,447)
(329,448)(330,435)(331,436)(332,437)(333,438)(334,439)(335,440)(336,441);;
s1 := ( 2, 7)( 3, 6)( 4, 5)( 9, 14)( 10, 13)( 11, 12)( 15, 22)( 16, 28)
( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 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, 99)( 86,105)( 87,104)( 88,103)( 89,102)( 90,101)( 91,100)( 92,106)
( 93,112)( 94,111)( 95,110)( 96,109)( 97,108)( 98,107)(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)(127,162)(128,168)(129,167)(130,166)
(131,165)(132,164)(133,163)(134,155)(135,161)(136,160)(137,159)(138,158)
(139,157)(140,156)(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)(239,302)(240,308)(241,307)(242,306)
(243,305)(244,304)(245,303)(246,295)(247,301)(248,300)(249,299)(250,298)
(251,297)(252,296)(253,309)(254,315)(255,314)(256,313)(257,312)(258,311)
(259,310)(260,316)(261,322)(262,321)(263,320)(264,319)(265,318)(266,317)
(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)(351,435)(352,441)(353,440)(354,439)
(355,438)(356,437)(357,436)(358,442)(359,448)(360,447)(361,446)(362,445)
(363,444)(364,443)(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)
(379,407)(380,413)(381,412)(382,411)(383,410)(384,409)(385,408)(386,414)
(387,420)(388,419)(389,418)(390,417)(391,416)(392,415);;
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)( 15,247)( 16,246)
( 17,252)( 18,251)( 19,250)( 20,249)( 21,248)( 22,240)( 23,239)( 24,245)
( 25,244)( 26,243)( 27,242)( 28,241)( 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)( 43,268)( 44,267)( 45,273)( 46,272)( 47,271)( 48,270)
( 49,269)( 50,275)( 51,274)( 52,280)( 53,279)( 54,278)( 55,277)( 56,276)
( 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)(127,359)(128,358)
(129,364)(130,363)(131,362)(132,361)(133,360)(134,352)(135,351)(136,357)
(137,356)(138,355)(139,354)(140,353)(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)(155,380)(156,379)(157,385)(158,384)(159,383)(160,382)
(161,381)(162,387)(163,386)(164,392)(165,391)(166,390)(167,389)(168,388)
(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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*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*s0*s1*s2*s1 ];;
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)( 29,141)( 30,142)( 31,143)
( 32,144)( 33,145)( 34,146)( 35,147)( 36,148)( 37,149)( 38,150)( 39,151)
( 40,152)( 41,153)( 42,154)( 43,155)( 44,156)( 45,157)( 46,158)( 47,159)
( 48,160)( 49,161)( 50,162)( 51,163)( 52,164)( 53,165)( 54,166)( 55,167)
( 56,168)( 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)( 85,204)( 86,205)( 87,206)
( 88,207)( 89,208)( 90,209)( 91,210)( 92,197)( 93,198)( 94,199)( 95,200)
( 96,201)( 97,202)( 98,203)( 99,218)(100,219)(101,220)(102,221)(103,222)
(104,223)(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)
(112,217)(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)(253,365)(254,366)(255,367)
(256,368)(257,369)(258,370)(259,371)(260,372)(261,373)(262,374)(263,375)
(264,376)(265,377)(266,378)(267,379)(268,380)(269,381)(270,382)(271,383)
(272,384)(273,385)(274,386)(275,387)(276,388)(277,389)(278,390)(279,391)
(280,392)(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)(309,428)(310,429)(311,430)
(312,431)(313,432)(314,433)(315,434)(316,421)(317,422)(318,423)(319,424)
(320,425)(321,426)(322,427)(323,442)(324,443)(325,444)(326,445)(327,446)
(328,447)(329,448)(330,435)(331,436)(332,437)(333,438)(334,439)(335,440)
(336,441);
s1 := Sym(448)!( 2, 7)( 3, 6)( 4, 5)( 9, 14)( 10, 13)( 11, 12)( 15, 22)
( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 30, 35)( 31, 34)
( 32, 33)( 37, 42)( 38, 41)( 39, 40)( 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, 99)( 86,105)( 87,104)( 88,103)( 89,102)( 90,101)( 91,100)
( 92,106)( 93,112)( 94,111)( 95,110)( 96,109)( 97,108)( 98,107)(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)(127,162)(128,168)(129,167)
(130,166)(131,165)(132,164)(133,163)(134,155)(135,161)(136,160)(137,159)
(138,158)(139,157)(140,156)(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)(239,302)(240,308)(241,307)
(242,306)(243,305)(244,304)(245,303)(246,295)(247,301)(248,300)(249,299)
(250,298)(251,297)(252,296)(253,309)(254,315)(255,314)(256,313)(257,312)
(258,311)(259,310)(260,316)(261,322)(262,321)(263,320)(264,319)(265,318)
(266,317)(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)(351,435)(352,441)(353,440)
(354,439)(355,438)(356,437)(357,436)(358,442)(359,448)(360,447)(361,446)
(362,445)(363,444)(364,443)(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)(379,407)(380,413)(381,412)(382,411)(383,410)(384,409)(385,408)
(386,414)(387,420)(388,419)(389,418)(390,417)(391,416)(392,415);
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)( 15,247)
( 16,246)( 17,252)( 18,251)( 19,250)( 20,249)( 21,248)( 22,240)( 23,239)
( 24,245)( 25,244)( 26,243)( 27,242)( 28,241)( 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)( 43,268)( 44,267)( 45,273)( 46,272)( 47,271)
( 48,270)( 49,269)( 50,275)( 51,274)( 52,280)( 53,279)( 54,278)( 55,277)
( 56,276)( 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)(127,359)
(128,358)(129,364)(130,363)(131,362)(132,361)(133,360)(134,352)(135,351)
(136,357)(137,356)(138,355)(139,354)(140,353)(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)(155,380)(156,379)(157,385)(158,384)(159,383)
(160,382)(161,381)(162,387)(163,386)(164,392)(165,391)(166,390)(167,389)
(168,388)(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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*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*s0*s1*s2*s1 >;
References : None.
to this polytope