Polytope of Type {4,112}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,112}*896a
Also Known As : {4,112|2}. if this polytope has another name.
Group : SmallGroup(896,1628)
Rank : 3
Schlafli Type : {4,112}
Number of vertices, edges, etc : 4, 224, 112
Order of s0s1s2 : 112
Order of s0s1s2s1 : 2
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, {2,112}*448
4-fold quotients : {4,28}*224, {2,56}*224
7-fold quotients : {4,16}*128a
8-fold quotients : {2,28}*112, {4,14}*112
14-fold quotients : {4,8}*64a, {2,16}*64
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}*1792c, {8,112}*1792d, {4,224}*1792a, {4,224}*1792b
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)
( 57,169)( 58,170)( 59,171)( 60,172)( 61,173)( 62,174)( 63,175)( 64,176)
( 65,177)( 66,178)( 67,179)( 68,180)( 69,181)( 70,182)( 71,183)( 72,184)
( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,190)( 79,191)( 80,192)
( 81,193)( 82,194)( 83,195)( 84,196)( 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)(253,365)(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)
(281,393)(282,394)(283,395)(284,396)(285,397)(286,398)(287,399)(288,400)
(289,401)(290,402)(291,403)(292,404)(293,405)(294,406)(295,407)(296,408)
(297,409)(298,410)(299,411)(300,412)(301,413)(302,414)(303,415)(304,416)
(305,417)(306,418)(307,419)(308,420)(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)( 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)(337,421)(338,427)
(339,426)(340,425)(341,424)(342,423)(343,422)(344,428)(345,434)(346,433)
(347,432)(348,431)(349,430)(350,429)(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)(365,393)(366,399)(367,398)(368,397)(369,396)(370,395)
(371,394)(372,400)(373,406)(374,405)(375,404)(376,403)(377,402)(378,401)
(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)( 15,247)( 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)( 29,254)( 30,253)( 31,259)( 32,258)
( 33,257)( 34,256)( 35,255)( 36,261)( 37,260)( 38,266)( 39,265)( 40,264)
( 41,263)( 42,262)( 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)( 85,324)( 86,323)( 87,329)( 88,328)
( 89,327)( 90,326)( 91,325)( 92,331)( 93,330)( 94,336)( 95,335)( 96,334)
( 97,333)( 98,332)( 99,310)(100,309)(101,315)(102,314)(103,313)(104,312)
(105,311)(106,317)(107,316)(108,322)(109,321)(110,320)(111,319)(112,318)
(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)(141,366)(142,365)(143,371)(144,370)
(145,369)(146,368)(147,367)(148,373)(149,372)(150,378)(151,377)(152,376)
(153,375)(154,374)(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)(197,436)(198,435)(199,441)(200,440)
(201,439)(202,438)(203,437)(204,443)(205,442)(206,448)(207,447)(208,446)
(209,445)(210,444)(211,422)(212,421)(213,427)(214,426)(215,425)(216,424)
(217,423)(218,429)(219,428)(220,434)(221,433)(222,432)(223,431)(224,430);;
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,
s0*s1*s2*s1*s0*s1*s2*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*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)( 29,141)( 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)( 57,169)( 58,170)( 59,171)( 60,172)( 61,173)( 62,174)( 63,175)
( 64,176)( 65,177)( 66,178)( 67,179)( 68,180)( 69,181)( 70,182)( 71,183)
( 72,184)( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,190)( 79,191)
( 80,192)( 81,193)( 82,194)( 83,195)( 84,196)( 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)(253,365)(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)(281,393)(282,394)(283,395)(284,396)(285,397)(286,398)(287,399)
(288,400)(289,401)(290,402)(291,403)(292,404)(293,405)(294,406)(295,407)
(296,408)(297,409)(298,410)(299,411)(300,412)(301,413)(302,414)(303,415)
(304,416)(305,417)(306,418)(307,419)(308,420)(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)( 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)(337,421)
(338,427)(339,426)(340,425)(341,424)(342,423)(343,422)(344,428)(345,434)
(346,433)(347,432)(348,431)(349,430)(350,429)(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)(365,393)(366,399)(367,398)(368,397)(369,396)
(370,395)(371,394)(372,400)(373,406)(374,405)(375,404)(376,403)(377,402)
(378,401)(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)( 15,247)
( 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)( 29,254)( 30,253)( 31,259)
( 32,258)( 33,257)( 34,256)( 35,255)( 36,261)( 37,260)( 38,266)( 39,265)
( 40,264)( 41,263)( 42,262)( 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)( 85,324)( 86,323)( 87,329)
( 88,328)( 89,327)( 90,326)( 91,325)( 92,331)( 93,330)( 94,336)( 95,335)
( 96,334)( 97,333)( 98,332)( 99,310)(100,309)(101,315)(102,314)(103,313)
(104,312)(105,311)(106,317)(107,316)(108,322)(109,321)(110,320)(111,319)
(112,318)(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)(141,366)(142,365)(143,371)
(144,370)(145,369)(146,368)(147,367)(148,373)(149,372)(150,378)(151,377)
(152,376)(153,375)(154,374)(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)(197,436)(198,435)(199,441)
(200,440)(201,439)(202,438)(203,437)(204,443)(205,442)(206,448)(207,447)
(208,446)(209,445)(210,444)(211,422)(212,421)(213,427)(214,426)(215,425)
(216,424)(217,423)(218,429)(219,428)(220,434)(221,433)(222,432)(223,431)
(224,430);
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,
s0*s1*s2*s1*s0*s1*s2*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*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