Polytope of Type {8,28}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,28}*896a
if this polytope has a name.
Group : SmallGroup(896,804)
Rank : 3
Schlafli Type : {8,28}
Number of vertices, edges, etc : 16, 224, 56
Order of s0s1s2 : 56
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,28,2} of size 1792
Vertex Figure Of :
{2,8,28} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,28}*448, {8,28}*448a, {8,28}*448b
4-fold quotients : {4,28}*224, {8,14}*224
7-fold quotients : {8,4}*128a
8-fold quotients : {2,28}*112, {4,14}*112
14-fold quotients : {8,4}*64a, {8,4}*64b, {4,4}*64
16-fold quotients : {2,14}*56
28-fold quotients : {4,4}*32, {8,2}*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 : {8,56}*1792a, {8,28}*1792a, {8,56}*1792c, {16,28}*1792a, {16,28}*1792b
Permutation Representation (GAP) :
s0 := ( 1,225)( 2,226)( 3,227)( 4,228)( 5,229)( 6,230)( 7,231)( 8,232)
( 9,233)( 10,234)( 11,235)( 12,236)( 13,237)( 14,238)( 15,239)( 16,240)
( 17,241)( 18,242)( 19,243)( 20,244)( 21,245)( 22,246)( 23,247)( 24,248)
( 25,249)( 26,250)( 27,251)( 28,252)( 29,260)( 30,261)( 31,262)( 32,263)
( 33,264)( 34,265)( 35,266)( 36,253)( 37,254)( 38,255)( 39,256)( 40,257)
( 41,258)( 42,259)( 43,274)( 44,275)( 45,276)( 46,277)( 47,278)( 48,279)
( 49,280)( 50,267)( 51,268)( 52,269)( 53,270)( 54,271)( 55,272)( 56,273)
( 57,295)( 58,296)( 59,297)( 60,298)( 61,299)( 62,300)( 63,301)( 64,302)
( 65,303)( 66,304)( 67,305)( 68,306)( 69,307)( 70,308)( 71,281)( 72,282)
( 73,283)( 74,284)( 75,285)( 76,286)( 77,287)( 78,288)( 79,289)( 80,290)
( 81,291)( 82,292)( 83,293)( 84,294)( 85,330)( 86,331)( 87,332)( 88,333)
( 89,334)( 90,335)( 91,336)( 92,323)( 93,324)( 94,325)( 95,326)( 96,327)
( 97,328)( 98,329)( 99,316)(100,317)(101,318)(102,319)(103,320)(104,321)
(105,322)(106,309)(107,310)(108,311)(109,312)(110,313)(111,314)(112,315)
(113,337)(114,338)(115,339)(116,340)(117,341)(118,342)(119,343)(120,344)
(121,345)(122,346)(123,347)(124,348)(125,349)(126,350)(127,351)(128,352)
(129,353)(130,354)(131,355)(132,356)(133,357)(134,358)(135,359)(136,360)
(137,361)(138,362)(139,363)(140,364)(141,372)(142,373)(143,374)(144,375)
(145,376)(146,377)(147,378)(148,365)(149,366)(150,367)(151,368)(152,369)
(153,370)(154,371)(155,386)(156,387)(157,388)(158,389)(159,390)(160,391)
(161,392)(162,379)(163,380)(164,381)(165,382)(166,383)(167,384)(168,385)
(169,407)(170,408)(171,409)(172,410)(173,411)(174,412)(175,413)(176,414)
(177,415)(178,416)(179,417)(180,418)(181,419)(182,420)(183,393)(184,394)
(185,395)(186,396)(187,397)(188,398)(189,399)(190,400)(191,401)(192,402)
(193,403)(194,404)(195,405)(196,406)(197,442)(198,443)(199,444)(200,445)
(201,446)(202,447)(203,448)(204,435)(205,436)(206,437)(207,438)(208,439)
(209,440)(210,441)(211,428)(212,429)(213,430)(214,431)(215,432)(216,433)
(217,434)(218,421)(219,422)(220,423)(221,424)(222,425)(223,426)(224,427);;
s1 := ( 2, 7)( 3, 6)( 4, 5)( 9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)
( 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)(127,155)(128,161)(129,160)(130,159)(131,158)(132,157)
(133,156)(134,162)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)
(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)(239,295)(240,301)(241,300)(242,299)(243,298)(244,297)
(245,296)(246,302)(247,308)(248,307)(249,306)(250,305)(251,304)(252,303)
(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)(267,323)(268,329)
(269,328)(270,327)(271,326)(272,325)(273,324)(274,330)(275,336)(276,335)
(277,334)(278,333)(279,332)(280,331)(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)(351,442)(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)
(365,400)(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)(379,414)(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 := ( 1,114)( 2,113)( 3,119)( 4,118)( 5,117)( 6,116)( 7,115)( 8,121)
( 9,120)( 10,126)( 11,125)( 12,124)( 13,123)( 14,122)( 15,128)( 16,127)
( 17,133)( 18,132)( 19,131)( 20,130)( 21,129)( 22,135)( 23,134)( 24,140)
( 25,139)( 26,138)( 27,137)( 28,136)( 29,142)( 30,141)( 31,147)( 32,146)
( 33,145)( 34,144)( 35,143)( 36,149)( 37,148)( 38,154)( 39,153)( 40,152)
( 41,151)( 42,150)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)( 48,158)
( 49,157)( 50,163)( 51,162)( 52,168)( 53,167)( 54,166)( 55,165)( 56,164)
( 57,177)( 58,176)( 59,182)( 60,181)( 61,180)( 62,179)( 63,178)( 64,170)
( 65,169)( 66,175)( 67,174)( 68,173)( 69,172)( 70,171)( 71,191)( 72,190)
( 73,196)( 74,195)( 75,194)( 76,193)( 77,192)( 78,184)( 79,183)( 80,189)
( 81,188)( 82,187)( 83,186)( 84,185)( 85,205)( 86,204)( 87,210)( 88,209)
( 89,208)( 90,207)( 91,206)( 92,198)( 93,197)( 94,203)( 95,202)( 96,201)
( 97,200)( 98,199)( 99,219)(100,218)(101,224)(102,223)(103,222)(104,221)
(105,220)(106,212)(107,211)(108,217)(109,216)(110,215)(111,214)(112,213)
(225,338)(226,337)(227,343)(228,342)(229,341)(230,340)(231,339)(232,345)
(233,344)(234,350)(235,349)(236,348)(237,347)(238,346)(239,352)(240,351)
(241,357)(242,356)(243,355)(244,354)(245,353)(246,359)(247,358)(248,364)
(249,363)(250,362)(251,361)(252,360)(253,366)(254,365)(255,371)(256,370)
(257,369)(258,368)(259,367)(260,373)(261,372)(262,378)(263,377)(264,376)
(265,375)(266,374)(267,380)(268,379)(269,385)(270,384)(271,383)(272,382)
(273,381)(274,387)(275,386)(276,392)(277,391)(278,390)(279,389)(280,388)
(281,401)(282,400)(283,406)(284,405)(285,404)(286,403)(287,402)(288,394)
(289,393)(290,399)(291,398)(292,397)(293,396)(294,395)(295,415)(296,414)
(297,420)(298,419)(299,418)(300,417)(301,416)(302,408)(303,407)(304,413)
(305,412)(306,411)(307,410)(308,409)(309,429)(310,428)(311,434)(312,433)
(313,432)(314,431)(315,430)(316,422)(317,421)(318,427)(319,426)(320,425)
(321,424)(322,423)(323,443)(324,442)(325,448)(326,447)(327,446)(328,445)
(329,444)(330,436)(331,435)(332,441)(333,440)(334,439)(335,438)(336,437);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(448)!( 1,225)( 2,226)( 3,227)( 4,228)( 5,229)( 6,230)( 7,231)
( 8,232)( 9,233)( 10,234)( 11,235)( 12,236)( 13,237)( 14,238)( 15,239)
( 16,240)( 17,241)( 18,242)( 19,243)( 20,244)( 21,245)( 22,246)( 23,247)
( 24,248)( 25,249)( 26,250)( 27,251)( 28,252)( 29,260)( 30,261)( 31,262)
( 32,263)( 33,264)( 34,265)( 35,266)( 36,253)( 37,254)( 38,255)( 39,256)
( 40,257)( 41,258)( 42,259)( 43,274)( 44,275)( 45,276)( 46,277)( 47,278)
( 48,279)( 49,280)( 50,267)( 51,268)( 52,269)( 53,270)( 54,271)( 55,272)
( 56,273)( 57,295)( 58,296)( 59,297)( 60,298)( 61,299)( 62,300)( 63,301)
( 64,302)( 65,303)( 66,304)( 67,305)( 68,306)( 69,307)( 70,308)( 71,281)
( 72,282)( 73,283)( 74,284)( 75,285)( 76,286)( 77,287)( 78,288)( 79,289)
( 80,290)( 81,291)( 82,292)( 83,293)( 84,294)( 85,330)( 86,331)( 87,332)
( 88,333)( 89,334)( 90,335)( 91,336)( 92,323)( 93,324)( 94,325)( 95,326)
( 96,327)( 97,328)( 98,329)( 99,316)(100,317)(101,318)(102,319)(103,320)
(104,321)(105,322)(106,309)(107,310)(108,311)(109,312)(110,313)(111,314)
(112,315)(113,337)(114,338)(115,339)(116,340)(117,341)(118,342)(119,343)
(120,344)(121,345)(122,346)(123,347)(124,348)(125,349)(126,350)(127,351)
(128,352)(129,353)(130,354)(131,355)(132,356)(133,357)(134,358)(135,359)
(136,360)(137,361)(138,362)(139,363)(140,364)(141,372)(142,373)(143,374)
(144,375)(145,376)(146,377)(147,378)(148,365)(149,366)(150,367)(151,368)
(152,369)(153,370)(154,371)(155,386)(156,387)(157,388)(158,389)(159,390)
(160,391)(161,392)(162,379)(163,380)(164,381)(165,382)(166,383)(167,384)
(168,385)(169,407)(170,408)(171,409)(172,410)(173,411)(174,412)(175,413)
(176,414)(177,415)(178,416)(179,417)(180,418)(181,419)(182,420)(183,393)
(184,394)(185,395)(186,396)(187,397)(188,398)(189,399)(190,400)(191,401)
(192,402)(193,403)(194,404)(195,405)(196,406)(197,442)(198,443)(199,444)
(200,445)(201,446)(202,447)(203,448)(204,435)(205,436)(206,437)(207,438)
(208,439)(209,440)(210,441)(211,428)(212,429)(213,430)(214,431)(215,432)
(216,433)(217,434)(218,421)(219,422)(220,423)(221,424)(222,425)(223,426)
(224,427);
s1 := Sym(448)!( 2, 7)( 3, 6)( 4, 5)( 9, 14)( 10, 13)( 11, 12)( 16, 21)
( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)
( 53, 54)( 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)(127,155)(128,161)(129,160)(130,159)(131,158)
(132,157)(133,156)(134,162)(135,168)(136,167)(137,166)(138,165)(139,164)
(140,163)(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)(239,295)(240,301)(241,300)(242,299)(243,298)
(244,297)(245,296)(246,302)(247,308)(248,307)(249,306)(250,305)(251,304)
(252,303)(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)(267,323)
(268,329)(269,328)(270,327)(271,326)(272,325)(273,324)(274,330)(275,336)
(276,335)(277,334)(278,333)(279,332)(280,331)(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)(351,442)(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)(365,400)(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)(379,414)
(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)!( 1,114)( 2,113)( 3,119)( 4,118)( 5,117)( 6,116)( 7,115)
( 8,121)( 9,120)( 10,126)( 11,125)( 12,124)( 13,123)( 14,122)( 15,128)
( 16,127)( 17,133)( 18,132)( 19,131)( 20,130)( 21,129)( 22,135)( 23,134)
( 24,140)( 25,139)( 26,138)( 27,137)( 28,136)( 29,142)( 30,141)( 31,147)
( 32,146)( 33,145)( 34,144)( 35,143)( 36,149)( 37,148)( 38,154)( 39,153)
( 40,152)( 41,151)( 42,150)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)
( 48,158)( 49,157)( 50,163)( 51,162)( 52,168)( 53,167)( 54,166)( 55,165)
( 56,164)( 57,177)( 58,176)( 59,182)( 60,181)( 61,180)( 62,179)( 63,178)
( 64,170)( 65,169)( 66,175)( 67,174)( 68,173)( 69,172)( 70,171)( 71,191)
( 72,190)( 73,196)( 74,195)( 75,194)( 76,193)( 77,192)( 78,184)( 79,183)
( 80,189)( 81,188)( 82,187)( 83,186)( 84,185)( 85,205)( 86,204)( 87,210)
( 88,209)( 89,208)( 90,207)( 91,206)( 92,198)( 93,197)( 94,203)( 95,202)
( 96,201)( 97,200)( 98,199)( 99,219)(100,218)(101,224)(102,223)(103,222)
(104,221)(105,220)(106,212)(107,211)(108,217)(109,216)(110,215)(111,214)
(112,213)(225,338)(226,337)(227,343)(228,342)(229,341)(230,340)(231,339)
(232,345)(233,344)(234,350)(235,349)(236,348)(237,347)(238,346)(239,352)
(240,351)(241,357)(242,356)(243,355)(244,354)(245,353)(246,359)(247,358)
(248,364)(249,363)(250,362)(251,361)(252,360)(253,366)(254,365)(255,371)
(256,370)(257,369)(258,368)(259,367)(260,373)(261,372)(262,378)(263,377)
(264,376)(265,375)(266,374)(267,380)(268,379)(269,385)(270,384)(271,383)
(272,382)(273,381)(274,387)(275,386)(276,392)(277,391)(278,390)(279,389)
(280,388)(281,401)(282,400)(283,406)(284,405)(285,404)(286,403)(287,402)
(288,394)(289,393)(290,399)(291,398)(292,397)(293,396)(294,395)(295,415)
(296,414)(297,420)(298,419)(299,418)(300,417)(301,416)(302,408)(303,407)
(304,413)(305,412)(306,411)(307,410)(308,409)(309,429)(310,428)(311,434)
(312,433)(313,432)(314,431)(315,430)(316,422)(317,421)(318,427)(319,426)
(320,425)(321,424)(322,423)(323,443)(324,442)(325,448)(326,447)(327,446)
(328,445)(329,444)(330,436)(331,435)(332,441)(333,440)(334,439)(335,438)
(336,437);
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*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 >;
References : None.
to this polytope