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# Polytope of Type {4,114}

Atlas Canonical Name : {4,114}*912b
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 : 114
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}*48c
38-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,228}*1824b, {4,228}*1824c, {4,114}*1824
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336)
(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)(350,352)
(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)(366,368)
(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)(382,384)
(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)(398,400)
(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)(414,416)
(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)(430,432)
(433,435)(434,436)(437,439)(438,440)(441,443)(442,444)(445,447)(446,448)
(449,451)(450,452)(453,455)(454,456);;
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*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336)(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)
(350,352)(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)
(366,368)(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)
(382,384)(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)
(398,400)(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)
(414,416)(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)
(430,432)(433,435)(434,436)(437,439)(438,440)(441,443)(442,444)(445,447)
(446,448)(449,451)(450,452)(453,455)(454,456);
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*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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