Polytope of Type {112,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {112,4}*896b
if this polytope has a name.
Group : SmallGroup(896,1640)
Rank : 3
Schlafli Type : {112,4}
Number of vertices, edges, etc : 112, 224, 4
Order of s0s1s2 : 112
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{112,4,2} of size 1792
Vertex Figure Of :
{2,112,4} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {56,4}*448a
4-fold quotients : {28,4}*224, {56,2}*224
7-fold quotients : {16,4}*128b
8-fold quotients : {28,2}*112, {14,4}*112
14-fold quotients : {8,4}*64a
16-fold quotients : {14,2}*56
28-fold quotients : {4,4}*32, {8,2}*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 : {112,4}*1792a, {112,8}*1792e, {112,8}*1792f
Permutation Representation (GAP) :
s0 := ( 1,225)( 2,231)( 3,230)( 4,229)( 5,228)( 6,227)( 7,226)( 8,232)
( 9,238)( 10,237)( 11,236)( 12,235)( 13,234)( 14,233)( 15,246)( 16,252)
( 17,251)( 18,250)( 19,249)( 20,248)( 21,247)( 22,239)( 23,245)( 24,244)
( 25,243)( 26,242)( 27,241)( 28,240)( 29,260)( 30,266)( 31,265)( 32,264)
( 33,263)( 34,262)( 35,261)( 36,253)( 37,259)( 38,258)( 39,257)( 40,256)
( 41,255)( 42,254)( 43,267)( 44,273)( 45,272)( 46,271)( 47,270)( 48,269)
( 49,268)( 50,274)( 51,280)( 52,279)( 53,278)( 54,277)( 55,276)( 56,275)
( 57,295)( 58,301)( 59,300)( 60,299)( 61,298)( 62,297)( 63,296)( 64,302)
( 65,308)( 66,307)( 67,306)( 68,305)( 69,304)( 70,303)( 71,281)( 72,287)
( 73,286)( 74,285)( 75,284)( 76,283)( 77,282)( 78,288)( 79,294)( 80,293)
( 81,292)( 82,291)( 83,290)( 84,289)( 85,330)( 86,336)( 87,335)( 88,334)
( 89,333)( 90,332)( 91,331)( 92,323)( 93,329)( 94,328)( 95,327)( 96,326)
( 97,325)( 98,324)( 99,316)(100,322)(101,321)(102,320)(103,319)(104,318)
(105,317)(106,309)(107,315)(108,314)(109,313)(110,312)(111,311)(112,310)
(113,337)(114,343)(115,342)(116,341)(117,340)(118,339)(119,338)(120,344)
(121,350)(122,349)(123,348)(124,347)(125,346)(126,345)(127,358)(128,364)
(129,363)(130,362)(131,361)(132,360)(133,359)(134,351)(135,357)(136,356)
(137,355)(138,354)(139,353)(140,352)(141,372)(142,378)(143,377)(144,376)
(145,375)(146,374)(147,373)(148,365)(149,371)(150,370)(151,369)(152,368)
(153,367)(154,366)(155,379)(156,385)(157,384)(158,383)(159,382)(160,381)
(161,380)(162,386)(163,392)(164,391)(165,390)(166,389)(167,388)(168,387)
(169,407)(170,413)(171,412)(172,411)(173,410)(174,409)(175,408)(176,414)
(177,420)(178,419)(179,418)(180,417)(181,416)(182,415)(183,393)(184,399)
(185,398)(186,397)(187,396)(188,395)(189,394)(190,400)(191,406)(192,405)
(193,404)(194,403)(195,402)(196,401)(197,442)(198,448)(199,447)(200,446)
(201,445)(202,444)(203,443)(204,435)(205,441)(206,440)(207,439)(208,438)
(209,437)(210,436)(211,428)(212,434)(213,433)(214,432)(215,431)(216,430)
(217,429)(218,421)(219,427)(220,426)(221,425)(222,424)(223,423)(224,422);;
s1 := ( 1, 2)( 3, 7)( 4, 6)( 8, 9)( 10, 14)( 11, 13)( 15, 23)( 16, 22)
( 17, 28)( 18, 27)( 19, 26)( 20, 25)( 21, 24)( 29, 30)( 31, 35)( 32, 34)
( 36, 37)( 38, 42)( 39, 41)( 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)
( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)( 92,107)
( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)(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)(127,163)(128,162)(129,168)(130,167)
(131,166)(132,165)(133,164)(134,156)(135,155)(136,161)(137,160)(138,159)
(139,158)(140,157)(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)(239,303)(240,302)(241,308)(242,307)
(243,306)(244,305)(245,304)(246,296)(247,295)(248,301)(249,300)(250,299)
(251,298)(252,297)(253,310)(254,309)(255,315)(256,314)(257,313)(258,312)
(259,311)(260,317)(261,316)(262,322)(263,321)(264,320)(265,319)(266,318)
(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)(351,436)(352,435)(353,441)(354,440)
(355,439)(356,438)(357,437)(358,443)(359,442)(360,448)(361,447)(362,446)
(363,445)(364,444)(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)
(379,408)(380,407)(381,413)(382,412)(383,411)(384,410)(385,409)(386,415)
(387,414)(388,420)(389,419)(390,418)(391,417)(392,416);;
s2 := ( 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);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*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*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*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)!( 1,225)( 2,231)( 3,230)( 4,229)( 5,228)( 6,227)( 7,226)
( 8,232)( 9,238)( 10,237)( 11,236)( 12,235)( 13,234)( 14,233)( 15,246)
( 16,252)( 17,251)( 18,250)( 19,249)( 20,248)( 21,247)( 22,239)( 23,245)
( 24,244)( 25,243)( 26,242)( 27,241)( 28,240)( 29,260)( 30,266)( 31,265)
( 32,264)( 33,263)( 34,262)( 35,261)( 36,253)( 37,259)( 38,258)( 39,257)
( 40,256)( 41,255)( 42,254)( 43,267)( 44,273)( 45,272)( 46,271)( 47,270)
( 48,269)( 49,268)( 50,274)( 51,280)( 52,279)( 53,278)( 54,277)( 55,276)
( 56,275)( 57,295)( 58,301)( 59,300)( 60,299)( 61,298)( 62,297)( 63,296)
( 64,302)( 65,308)( 66,307)( 67,306)( 68,305)( 69,304)( 70,303)( 71,281)
( 72,287)( 73,286)( 74,285)( 75,284)( 76,283)( 77,282)( 78,288)( 79,294)
( 80,293)( 81,292)( 82,291)( 83,290)( 84,289)( 85,330)( 86,336)( 87,335)
( 88,334)( 89,333)( 90,332)( 91,331)( 92,323)( 93,329)( 94,328)( 95,327)
( 96,326)( 97,325)( 98,324)( 99,316)(100,322)(101,321)(102,320)(103,319)
(104,318)(105,317)(106,309)(107,315)(108,314)(109,313)(110,312)(111,311)
(112,310)(113,337)(114,343)(115,342)(116,341)(117,340)(118,339)(119,338)
(120,344)(121,350)(122,349)(123,348)(124,347)(125,346)(126,345)(127,358)
(128,364)(129,363)(130,362)(131,361)(132,360)(133,359)(134,351)(135,357)
(136,356)(137,355)(138,354)(139,353)(140,352)(141,372)(142,378)(143,377)
(144,376)(145,375)(146,374)(147,373)(148,365)(149,371)(150,370)(151,369)
(152,368)(153,367)(154,366)(155,379)(156,385)(157,384)(158,383)(159,382)
(160,381)(161,380)(162,386)(163,392)(164,391)(165,390)(166,389)(167,388)
(168,387)(169,407)(170,413)(171,412)(172,411)(173,410)(174,409)(175,408)
(176,414)(177,420)(178,419)(179,418)(180,417)(181,416)(182,415)(183,393)
(184,399)(185,398)(186,397)(187,396)(188,395)(189,394)(190,400)(191,406)
(192,405)(193,404)(194,403)(195,402)(196,401)(197,442)(198,448)(199,447)
(200,446)(201,445)(202,444)(203,443)(204,435)(205,441)(206,440)(207,439)
(208,438)(209,437)(210,436)(211,428)(212,434)(213,433)(214,432)(215,431)
(216,430)(217,429)(218,421)(219,427)(220,426)(221,425)(222,424)(223,423)
(224,422);
s1 := Sym(448)!( 1, 2)( 3, 7)( 4, 6)( 8, 9)( 10, 14)( 11, 13)( 15, 23)
( 16, 22)( 17, 28)( 18, 27)( 19, 26)( 20, 25)( 21, 24)( 29, 30)( 31, 35)
( 32, 34)( 36, 37)( 38, 42)( 39, 41)( 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)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)
( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)(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)(127,163)(128,162)(129,168)
(130,167)(131,166)(132,165)(133,164)(134,156)(135,155)(136,161)(137,160)
(138,159)(139,158)(140,157)(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)(239,303)(240,302)(241,308)
(242,307)(243,306)(244,305)(245,304)(246,296)(247,295)(248,301)(249,300)
(250,299)(251,298)(252,297)(253,310)(254,309)(255,315)(256,314)(257,313)
(258,312)(259,311)(260,317)(261,316)(262,322)(263,321)(264,320)(265,319)
(266,318)(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)(351,436)(352,435)(353,441)
(354,440)(355,439)(356,438)(357,437)(358,443)(359,442)(360,448)(361,447)
(362,446)(363,445)(364,444)(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)(379,408)(380,407)(381,413)(382,412)(383,411)(384,410)(385,409)
(386,415)(387,414)(388,420)(389,419)(390,418)(391,417)(392,416);
s2 := 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);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*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*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*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