Polytope of Type {4,114}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,114}*912c
if this polytope has a name.
Group : SmallGroup(912,209)
Rank : 3
Schlafli Type : {4,114}
Number of vertices, edges, etc : 4, 228, 114
Order of s0s1s2 : 57
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,114,2} of size 1824
Vertex Figure Of :
{2,4,114} of size 1824
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,57}*456
19-fold quotients : {4,6}*48b
38-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,114}*1824
Permutation Representation (GAP) :
s0 := ( 1,231)( 2,232)( 3,229)( 4,230)( 5,235)( 6,236)( 7,233)( 8,234)
( 9,239)( 10,240)( 11,237)( 12,238)( 13,243)( 14,244)( 15,241)( 16,242)
( 17,247)( 18,248)( 19,245)( 20,246)( 21,251)( 22,252)( 23,249)( 24,250)
( 25,255)( 26,256)( 27,253)( 28,254)( 29,259)( 30,260)( 31,257)( 32,258)
( 33,263)( 34,264)( 35,261)( 36,262)( 37,267)( 38,268)( 39,265)( 40,266)
( 41,271)( 42,272)( 43,269)( 44,270)( 45,275)( 46,276)( 47,273)( 48,274)
( 49,279)( 50,280)( 51,277)( 52,278)( 53,283)( 54,284)( 55,281)( 56,282)
( 57,287)( 58,288)( 59,285)( 60,286)( 61,291)( 62,292)( 63,289)( 64,290)
( 65,295)( 66,296)( 67,293)( 68,294)( 69,299)( 70,300)( 71,297)( 72,298)
( 73,303)( 74,304)( 75,301)( 76,302)( 77,307)( 78,308)( 79,305)( 80,306)
( 81,311)( 82,312)( 83,309)( 84,310)( 85,315)( 86,316)( 87,313)( 88,314)
( 89,319)( 90,320)( 91,317)( 92,318)( 93,323)( 94,324)( 95,321)( 96,322)
( 97,327)( 98,328)( 99,325)(100,326)(101,331)(102,332)(103,329)(104,330)
(105,335)(106,336)(107,333)(108,334)(109,339)(110,340)(111,337)(112,338)
(113,343)(114,344)(115,341)(116,342)(117,347)(118,348)(119,345)(120,346)
(121,351)(122,352)(123,349)(124,350)(125,355)(126,356)(127,353)(128,354)
(129,359)(130,360)(131,357)(132,358)(133,363)(134,364)(135,361)(136,362)
(137,367)(138,368)(139,365)(140,366)(141,371)(142,372)(143,369)(144,370)
(145,375)(146,376)(147,373)(148,374)(149,379)(150,380)(151,377)(152,378)
(153,383)(154,384)(155,381)(156,382)(157,387)(158,388)(159,385)(160,386)
(161,391)(162,392)(163,389)(164,390)(165,395)(166,396)(167,393)(168,394)
(169,399)(170,400)(171,397)(172,398)(173,403)(174,404)(175,401)(176,402)
(177,407)(178,408)(179,405)(180,406)(181,411)(182,412)(183,409)(184,410)
(185,415)(186,416)(187,413)(188,414)(189,419)(190,420)(191,417)(192,418)
(193,423)(194,424)(195,421)(196,422)(197,427)(198,428)(199,425)(200,426)
(201,431)(202,432)(203,429)(204,430)(205,435)(206,436)(207,433)(208,434)
(209,439)(210,440)(211,437)(212,438)(213,443)(214,444)(215,441)(216,442)
(217,447)(218,448)(219,445)(220,446)(221,451)(222,452)(223,449)(224,450)
(225,455)(226,456)(227,453)(228,454);;
s1 := ( 2, 3)( 5, 73)( 6, 75)( 7, 74)( 8, 76)( 9, 69)( 10, 71)( 11, 70)
( 12, 72)( 13, 65)( 14, 67)( 15, 66)( 16, 68)( 17, 61)( 18, 63)( 19, 62)
( 20, 64)( 21, 57)( 22, 59)( 23, 58)( 24, 60)( 25, 53)( 26, 55)( 27, 54)
( 28, 56)( 29, 49)( 30, 51)( 31, 50)( 32, 52)( 33, 45)( 34, 47)( 35, 46)
( 36, 48)( 37, 41)( 38, 43)( 39, 42)( 40, 44)( 77,153)( 78,155)( 79,154)
( 80,156)( 81,225)( 82,227)( 83,226)( 84,228)( 85,221)( 86,223)( 87,222)
( 88,224)( 89,217)( 90,219)( 91,218)( 92,220)( 93,213)( 94,215)( 95,214)
( 96,216)( 97,209)( 98,211)( 99,210)(100,212)(101,205)(102,207)(103,206)
(104,208)(105,201)(106,203)(107,202)(108,204)(109,197)(110,199)(111,198)
(112,200)(113,193)(114,195)(115,194)(116,196)(117,189)(118,191)(119,190)
(120,192)(121,185)(122,187)(123,186)(124,188)(125,181)(126,183)(127,182)
(128,184)(129,177)(130,179)(131,178)(132,180)(133,173)(134,175)(135,174)
(136,176)(137,169)(138,171)(139,170)(140,172)(141,165)(142,167)(143,166)
(144,168)(145,161)(146,163)(147,162)(148,164)(149,157)(150,159)(151,158)
(152,160)(230,231)(233,301)(234,303)(235,302)(236,304)(237,297)(238,299)
(239,298)(240,300)(241,293)(242,295)(243,294)(244,296)(245,289)(246,291)
(247,290)(248,292)(249,285)(250,287)(251,286)(252,288)(253,281)(254,283)
(255,282)(256,284)(257,277)(258,279)(259,278)(260,280)(261,273)(262,275)
(263,274)(264,276)(265,269)(266,271)(267,270)(268,272)(305,381)(306,383)
(307,382)(308,384)(309,453)(310,455)(311,454)(312,456)(313,449)(314,451)
(315,450)(316,452)(317,445)(318,447)(319,446)(320,448)(321,441)(322,443)
(323,442)(324,444)(325,437)(326,439)(327,438)(328,440)(329,433)(330,435)
(331,434)(332,436)(333,429)(334,431)(335,430)(336,432)(337,425)(338,427)
(339,426)(340,428)(341,421)(342,423)(343,422)(344,424)(345,417)(346,419)
(347,418)(348,420)(349,413)(350,415)(351,414)(352,416)(353,409)(354,411)
(355,410)(356,412)(357,405)(358,407)(359,406)(360,408)(361,401)(362,403)
(363,402)(364,404)(365,397)(366,399)(367,398)(368,400)(369,393)(370,395)
(371,394)(372,396)(373,389)(374,391)(375,390)(376,392)(377,385)(378,387)
(379,386)(380,388);;
s2 := ( 1,385)( 2,388)( 3,387)( 4,386)( 5,381)( 6,384)( 7,383)( 8,382)
( 9,453)( 10,456)( 11,455)( 12,454)( 13,449)( 14,452)( 15,451)( 16,450)
( 17,445)( 18,448)( 19,447)( 20,446)( 21,441)( 22,444)( 23,443)( 24,442)
( 25,437)( 26,440)( 27,439)( 28,438)( 29,433)( 30,436)( 31,435)( 32,434)
( 33,429)( 34,432)( 35,431)( 36,430)( 37,425)( 38,428)( 39,427)( 40,426)
( 41,421)( 42,424)( 43,423)( 44,422)( 45,417)( 46,420)( 47,419)( 48,418)
( 49,413)( 50,416)( 51,415)( 52,414)( 53,409)( 54,412)( 55,411)( 56,410)
( 57,405)( 58,408)( 59,407)( 60,406)( 61,401)( 62,404)( 63,403)( 64,402)
( 65,397)( 66,400)( 67,399)( 68,398)( 69,393)( 70,396)( 71,395)( 72,394)
( 73,389)( 74,392)( 75,391)( 76,390)( 77,309)( 78,312)( 79,311)( 80,310)
( 81,305)( 82,308)( 83,307)( 84,306)( 85,377)( 86,380)( 87,379)( 88,378)
( 89,373)( 90,376)( 91,375)( 92,374)( 93,369)( 94,372)( 95,371)( 96,370)
( 97,365)( 98,368)( 99,367)(100,366)(101,361)(102,364)(103,363)(104,362)
(105,357)(106,360)(107,359)(108,358)(109,353)(110,356)(111,355)(112,354)
(113,349)(114,352)(115,351)(116,350)(117,345)(118,348)(119,347)(120,346)
(121,341)(122,344)(123,343)(124,342)(125,337)(126,340)(127,339)(128,338)
(129,333)(130,336)(131,335)(132,334)(133,329)(134,332)(135,331)(136,330)
(137,325)(138,328)(139,327)(140,326)(141,321)(142,324)(143,323)(144,322)
(145,317)(146,320)(147,319)(148,318)(149,313)(150,316)(151,315)(152,314)
(153,233)(154,236)(155,235)(156,234)(157,229)(158,232)(159,231)(160,230)
(161,301)(162,304)(163,303)(164,302)(165,297)(166,300)(167,299)(168,298)
(169,293)(170,296)(171,295)(172,294)(173,289)(174,292)(175,291)(176,290)
(177,285)(178,288)(179,287)(180,286)(181,281)(182,284)(183,283)(184,282)
(185,277)(186,280)(187,279)(188,278)(189,273)(190,276)(191,275)(192,274)
(193,269)(194,272)(195,271)(196,270)(197,265)(198,268)(199,267)(200,266)
(201,261)(202,264)(203,263)(204,262)(205,257)(206,260)(207,259)(208,258)
(209,253)(210,256)(211,255)(212,254)(213,249)(214,252)(215,251)(216,250)
(217,245)(218,248)(219,247)(220,246)(221,241)(222,244)(223,243)(224,242)
(225,237)(226,240)(227,239)(228,238);;
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*s2*s1*s0*s1*s2*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*s0*s2*s1*s0*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(456)!( 1,231)( 2,232)( 3,229)( 4,230)( 5,235)( 6,236)( 7,233)
( 8,234)( 9,239)( 10,240)( 11,237)( 12,238)( 13,243)( 14,244)( 15,241)
( 16,242)( 17,247)( 18,248)( 19,245)( 20,246)( 21,251)( 22,252)( 23,249)
( 24,250)( 25,255)( 26,256)( 27,253)( 28,254)( 29,259)( 30,260)( 31,257)
( 32,258)( 33,263)( 34,264)( 35,261)( 36,262)( 37,267)( 38,268)( 39,265)
( 40,266)( 41,271)( 42,272)( 43,269)( 44,270)( 45,275)( 46,276)( 47,273)
( 48,274)( 49,279)( 50,280)( 51,277)( 52,278)( 53,283)( 54,284)( 55,281)
( 56,282)( 57,287)( 58,288)( 59,285)( 60,286)( 61,291)( 62,292)( 63,289)
( 64,290)( 65,295)( 66,296)( 67,293)( 68,294)( 69,299)( 70,300)( 71,297)
( 72,298)( 73,303)( 74,304)( 75,301)( 76,302)( 77,307)( 78,308)( 79,305)
( 80,306)( 81,311)( 82,312)( 83,309)( 84,310)( 85,315)( 86,316)( 87,313)
( 88,314)( 89,319)( 90,320)( 91,317)( 92,318)( 93,323)( 94,324)( 95,321)
( 96,322)( 97,327)( 98,328)( 99,325)(100,326)(101,331)(102,332)(103,329)
(104,330)(105,335)(106,336)(107,333)(108,334)(109,339)(110,340)(111,337)
(112,338)(113,343)(114,344)(115,341)(116,342)(117,347)(118,348)(119,345)
(120,346)(121,351)(122,352)(123,349)(124,350)(125,355)(126,356)(127,353)
(128,354)(129,359)(130,360)(131,357)(132,358)(133,363)(134,364)(135,361)
(136,362)(137,367)(138,368)(139,365)(140,366)(141,371)(142,372)(143,369)
(144,370)(145,375)(146,376)(147,373)(148,374)(149,379)(150,380)(151,377)
(152,378)(153,383)(154,384)(155,381)(156,382)(157,387)(158,388)(159,385)
(160,386)(161,391)(162,392)(163,389)(164,390)(165,395)(166,396)(167,393)
(168,394)(169,399)(170,400)(171,397)(172,398)(173,403)(174,404)(175,401)
(176,402)(177,407)(178,408)(179,405)(180,406)(181,411)(182,412)(183,409)
(184,410)(185,415)(186,416)(187,413)(188,414)(189,419)(190,420)(191,417)
(192,418)(193,423)(194,424)(195,421)(196,422)(197,427)(198,428)(199,425)
(200,426)(201,431)(202,432)(203,429)(204,430)(205,435)(206,436)(207,433)
(208,434)(209,439)(210,440)(211,437)(212,438)(213,443)(214,444)(215,441)
(216,442)(217,447)(218,448)(219,445)(220,446)(221,451)(222,452)(223,449)
(224,450)(225,455)(226,456)(227,453)(228,454);
s1 := Sym(456)!( 2, 3)( 5, 73)( 6, 75)( 7, 74)( 8, 76)( 9, 69)( 10, 71)
( 11, 70)( 12, 72)( 13, 65)( 14, 67)( 15, 66)( 16, 68)( 17, 61)( 18, 63)
( 19, 62)( 20, 64)( 21, 57)( 22, 59)( 23, 58)( 24, 60)( 25, 53)( 26, 55)
( 27, 54)( 28, 56)( 29, 49)( 30, 51)( 31, 50)( 32, 52)( 33, 45)( 34, 47)
( 35, 46)( 36, 48)( 37, 41)( 38, 43)( 39, 42)( 40, 44)( 77,153)( 78,155)
( 79,154)( 80,156)( 81,225)( 82,227)( 83,226)( 84,228)( 85,221)( 86,223)
( 87,222)( 88,224)( 89,217)( 90,219)( 91,218)( 92,220)( 93,213)( 94,215)
( 95,214)( 96,216)( 97,209)( 98,211)( 99,210)(100,212)(101,205)(102,207)
(103,206)(104,208)(105,201)(106,203)(107,202)(108,204)(109,197)(110,199)
(111,198)(112,200)(113,193)(114,195)(115,194)(116,196)(117,189)(118,191)
(119,190)(120,192)(121,185)(122,187)(123,186)(124,188)(125,181)(126,183)
(127,182)(128,184)(129,177)(130,179)(131,178)(132,180)(133,173)(134,175)
(135,174)(136,176)(137,169)(138,171)(139,170)(140,172)(141,165)(142,167)
(143,166)(144,168)(145,161)(146,163)(147,162)(148,164)(149,157)(150,159)
(151,158)(152,160)(230,231)(233,301)(234,303)(235,302)(236,304)(237,297)
(238,299)(239,298)(240,300)(241,293)(242,295)(243,294)(244,296)(245,289)
(246,291)(247,290)(248,292)(249,285)(250,287)(251,286)(252,288)(253,281)
(254,283)(255,282)(256,284)(257,277)(258,279)(259,278)(260,280)(261,273)
(262,275)(263,274)(264,276)(265,269)(266,271)(267,270)(268,272)(305,381)
(306,383)(307,382)(308,384)(309,453)(310,455)(311,454)(312,456)(313,449)
(314,451)(315,450)(316,452)(317,445)(318,447)(319,446)(320,448)(321,441)
(322,443)(323,442)(324,444)(325,437)(326,439)(327,438)(328,440)(329,433)
(330,435)(331,434)(332,436)(333,429)(334,431)(335,430)(336,432)(337,425)
(338,427)(339,426)(340,428)(341,421)(342,423)(343,422)(344,424)(345,417)
(346,419)(347,418)(348,420)(349,413)(350,415)(351,414)(352,416)(353,409)
(354,411)(355,410)(356,412)(357,405)(358,407)(359,406)(360,408)(361,401)
(362,403)(363,402)(364,404)(365,397)(366,399)(367,398)(368,400)(369,393)
(370,395)(371,394)(372,396)(373,389)(374,391)(375,390)(376,392)(377,385)
(378,387)(379,386)(380,388);
s2 := Sym(456)!( 1,385)( 2,388)( 3,387)( 4,386)( 5,381)( 6,384)( 7,383)
( 8,382)( 9,453)( 10,456)( 11,455)( 12,454)( 13,449)( 14,452)( 15,451)
( 16,450)( 17,445)( 18,448)( 19,447)( 20,446)( 21,441)( 22,444)( 23,443)
( 24,442)( 25,437)( 26,440)( 27,439)( 28,438)( 29,433)( 30,436)( 31,435)
( 32,434)( 33,429)( 34,432)( 35,431)( 36,430)( 37,425)( 38,428)( 39,427)
( 40,426)( 41,421)( 42,424)( 43,423)( 44,422)( 45,417)( 46,420)( 47,419)
( 48,418)( 49,413)( 50,416)( 51,415)( 52,414)( 53,409)( 54,412)( 55,411)
( 56,410)( 57,405)( 58,408)( 59,407)( 60,406)( 61,401)( 62,404)( 63,403)
( 64,402)( 65,397)( 66,400)( 67,399)( 68,398)( 69,393)( 70,396)( 71,395)
( 72,394)( 73,389)( 74,392)( 75,391)( 76,390)( 77,309)( 78,312)( 79,311)
( 80,310)( 81,305)( 82,308)( 83,307)( 84,306)( 85,377)( 86,380)( 87,379)
( 88,378)( 89,373)( 90,376)( 91,375)( 92,374)( 93,369)( 94,372)( 95,371)
( 96,370)( 97,365)( 98,368)( 99,367)(100,366)(101,361)(102,364)(103,363)
(104,362)(105,357)(106,360)(107,359)(108,358)(109,353)(110,356)(111,355)
(112,354)(113,349)(114,352)(115,351)(116,350)(117,345)(118,348)(119,347)
(120,346)(121,341)(122,344)(123,343)(124,342)(125,337)(126,340)(127,339)
(128,338)(129,333)(130,336)(131,335)(132,334)(133,329)(134,332)(135,331)
(136,330)(137,325)(138,328)(139,327)(140,326)(141,321)(142,324)(143,323)
(144,322)(145,317)(146,320)(147,319)(148,318)(149,313)(150,316)(151,315)
(152,314)(153,233)(154,236)(155,235)(156,234)(157,229)(158,232)(159,231)
(160,230)(161,301)(162,304)(163,303)(164,302)(165,297)(166,300)(167,299)
(168,298)(169,293)(170,296)(171,295)(172,294)(173,289)(174,292)(175,291)
(176,290)(177,285)(178,288)(179,287)(180,286)(181,281)(182,284)(183,283)
(184,282)(185,277)(186,280)(187,279)(188,278)(189,273)(190,276)(191,275)
(192,274)(193,269)(194,272)(195,271)(196,270)(197,265)(198,268)(199,267)
(200,266)(201,261)(202,264)(203,263)(204,262)(205,257)(206,260)(207,259)
(208,258)(209,253)(210,256)(211,255)(212,254)(213,249)(214,252)(215,251)
(216,250)(217,245)(218,248)(219,247)(220,246)(221,241)(222,244)(223,243)
(224,242)(225,237)(226,240)(227,239)(228,238);
poly := sub<Sym(456)|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*s2*s1*s0*s1*s2*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*s0*s2*s1*s0*s2*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 >;
References : None.
to this polytope