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Polytope of Type {4,32}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,32}*512b
if this polytope has a name.
Group : SmallGroup(512,58378)
Rank : 3
Schlafli Type : {4,32}
Number of vertices, edges, etc : 8, 128, 64
Order of s0s1s2 : 32
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,16}*256a
   4-fold quotients : {4,8}*128a, {4,16}*128a, {4,16}*128b
   8-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64, {2,16}*64
   16-fold quotients : {4,4}*32, {2,8}*32
   32-fold quotients : {2,4}*16, {4,2}*16
   64-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)(  8, 72)
(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)( 16, 80)
( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 87)( 24, 88)
( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)( 32, 96)
( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)( 40,100)
( 41,109)( 42,110)( 43,111)( 44,112)( 45,105)( 46,106)( 47,107)( 48,108)
( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)
( 57,125)( 58,126)( 59,127)( 60,128)( 61,121)( 62,122)( 63,123)( 64,124)
(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)(136,200)
(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)(144,208)
(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)(152,216)
(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)(160,224)
(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)(168,228)
(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)(176,236)
(177,245)(178,246)(179,247)(180,248)(181,241)(182,242)(183,243)(184,244)
(185,253)(186,254)(187,255)(188,256)(189,249)(190,250)(191,251)(192,252)
(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)(264,328)
(265,329)(266,330)(267,331)(268,332)(269,333)(270,334)(271,335)(272,336)
(273,337)(274,338)(275,339)(276,340)(277,341)(278,342)(279,343)(280,344)
(281,345)(282,346)(283,347)(284,348)(285,349)(286,350)(287,351)(288,352)
(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)(296,356)
(297,365)(298,366)(299,367)(300,368)(301,361)(302,362)(303,363)(304,364)
(305,373)(306,374)(307,375)(308,376)(309,369)(310,370)(311,371)(312,372)
(313,381)(314,382)(315,383)(316,384)(317,377)(318,378)(319,379)(320,380)
(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)(392,456)
(393,457)(394,458)(395,459)(396,460)(397,461)(398,462)(399,463)(400,464)
(401,465)(402,466)(403,467)(404,468)(405,469)(406,470)(407,471)(408,472)
(409,473)(410,474)(411,475)(412,476)(413,477)(414,478)(415,479)(416,480)
(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)(424,484)
(425,493)(426,494)(427,495)(428,496)(429,489)(430,490)(431,491)(432,492)
(433,501)(434,502)(435,503)(436,504)(437,497)(438,498)(439,499)(440,500)
(441,509)(442,510)(443,511)(444,512)(445,505)(446,506)(447,507)(448,508);;
s1 := (  1,257)(  2,258)(  3,260)(  4,259)(  5,262)(  6,261)(  7,263)(  8,264)
(  9,267)( 10,268)( 11,265)( 12,266)( 13,272)( 14,271)( 15,270)( 16,269)
( 17,273)( 18,274)( 19,276)( 20,275)( 21,278)( 22,277)( 23,279)( 24,280)
( 25,283)( 26,284)( 27,281)( 28,282)( 29,288)( 30,287)( 31,286)( 32,285)
( 33,297)( 34,298)( 35,300)( 36,299)( 37,302)( 38,301)( 39,303)( 40,304)
( 41,289)( 42,290)( 43,292)( 44,291)( 45,294)( 46,293)( 47,295)( 48,296)
( 49,313)( 50,314)( 51,316)( 52,315)( 53,318)( 54,317)( 55,319)( 56,320)
( 57,305)( 58,306)( 59,308)( 60,307)( 61,310)( 62,309)( 63,311)( 64,312)
( 65,337)( 66,338)( 67,340)( 68,339)( 69,342)( 70,341)( 71,343)( 72,344)
( 73,347)( 74,348)( 75,345)( 76,346)( 77,352)( 78,351)( 79,350)( 80,349)
( 81,321)( 82,322)( 83,324)( 84,323)( 85,326)( 86,325)( 87,327)( 88,328)
( 89,331)( 90,332)( 91,329)( 92,330)( 93,336)( 94,335)( 95,334)( 96,333)
( 97,378)( 98,377)( 99,379)(100,380)(101,381)(102,382)(103,384)(104,383)
(105,370)(106,369)(107,371)(108,372)(109,373)(110,374)(111,376)(112,375)
(113,362)(114,361)(115,363)(116,364)(117,365)(118,366)(119,368)(120,367)
(121,354)(122,353)(123,355)(124,356)(125,357)(126,358)(127,360)(128,359)
(129,417)(130,418)(131,420)(132,419)(133,422)(134,421)(135,423)(136,424)
(137,427)(138,428)(139,425)(140,426)(141,432)(142,431)(143,430)(144,429)
(145,433)(146,434)(147,436)(148,435)(149,438)(150,437)(151,439)(152,440)
(153,443)(154,444)(155,441)(156,442)(157,448)(158,447)(159,446)(160,445)
(161,385)(162,386)(163,388)(164,387)(165,390)(166,389)(167,391)(168,392)
(169,395)(170,396)(171,393)(172,394)(173,400)(174,399)(175,398)(176,397)
(177,401)(178,402)(179,404)(180,403)(181,406)(182,405)(183,407)(184,408)
(185,411)(186,412)(187,409)(188,410)(189,416)(190,415)(191,414)(192,413)
(193,501)(194,502)(195,504)(196,503)(197,498)(198,497)(199,499)(200,500)
(201,511)(202,512)(203,509)(204,510)(205,508)(206,507)(207,506)(208,505)
(209,485)(210,486)(211,488)(212,487)(213,482)(214,481)(215,483)(216,484)
(217,495)(218,496)(219,493)(220,494)(221,492)(222,491)(223,490)(224,489)
(225,470)(226,469)(227,471)(228,472)(229,465)(230,466)(231,468)(232,467)
(233,480)(234,479)(235,478)(236,477)(237,475)(238,476)(239,473)(240,474)
(241,454)(242,453)(243,455)(244,456)(245,449)(246,450)(247,452)(248,451)
(249,464)(250,463)(251,462)(252,461)(253,459)(254,460)(255,457)(256,458);;
s2 := (  1,129)(  2,130)(  3,132)(  4,131)(  5,133)(  6,134)(  7,136)(  8,135)
(  9,139)( 10,140)( 11,137)( 12,138)( 13,143)( 14,144)( 15,141)( 16,142)
( 17,150)( 18,149)( 19,151)( 20,152)( 21,146)( 22,145)( 23,147)( 24,148)
( 25,160)( 26,159)( 27,158)( 28,157)( 29,156)( 30,155)( 31,154)( 32,153)
( 33,169)( 34,170)( 35,172)( 36,171)( 37,173)( 38,174)( 39,176)( 40,175)
( 41,161)( 42,162)( 43,164)( 44,163)( 45,165)( 46,166)( 47,168)( 48,167)
( 49,190)( 50,189)( 51,191)( 52,192)( 53,186)( 54,185)( 55,187)( 56,188)
( 57,182)( 58,181)( 59,183)( 60,184)( 61,178)( 62,177)( 63,179)( 64,180)
( 65,193)( 66,194)( 67,196)( 68,195)( 69,197)( 70,198)( 71,200)( 72,199)
( 73,203)( 74,204)( 75,201)( 76,202)( 77,207)( 78,208)( 79,205)( 80,206)
( 81,214)( 82,213)( 83,215)( 84,216)( 85,210)( 86,209)( 87,211)( 88,212)
( 89,224)( 90,223)( 91,222)( 92,221)( 93,220)( 94,219)( 95,218)( 96,217)
( 97,233)( 98,234)( 99,236)(100,235)(101,237)(102,238)(103,240)(104,239)
(105,225)(106,226)(107,228)(108,227)(109,229)(110,230)(111,232)(112,231)
(113,254)(114,253)(115,255)(116,256)(117,250)(118,249)(119,251)(120,252)
(121,246)(122,245)(123,247)(124,248)(125,242)(126,241)(127,243)(128,244)
(257,385)(258,386)(259,388)(260,387)(261,389)(262,390)(263,392)(264,391)
(265,395)(266,396)(267,393)(268,394)(269,399)(270,400)(271,397)(272,398)
(273,406)(274,405)(275,407)(276,408)(277,402)(278,401)(279,403)(280,404)
(281,416)(282,415)(283,414)(284,413)(285,412)(286,411)(287,410)(288,409)
(289,425)(290,426)(291,428)(292,427)(293,429)(294,430)(295,432)(296,431)
(297,417)(298,418)(299,420)(300,419)(301,421)(302,422)(303,424)(304,423)
(305,446)(306,445)(307,447)(308,448)(309,442)(310,441)(311,443)(312,444)
(313,438)(314,437)(315,439)(316,440)(317,434)(318,433)(319,435)(320,436)
(321,449)(322,450)(323,452)(324,451)(325,453)(326,454)(327,456)(328,455)
(329,459)(330,460)(331,457)(332,458)(333,463)(334,464)(335,461)(336,462)
(337,470)(338,469)(339,471)(340,472)(341,466)(342,465)(343,467)(344,468)
(345,480)(346,479)(347,478)(348,477)(349,476)(350,475)(351,474)(352,473)
(353,489)(354,490)(355,492)(356,491)(357,493)(358,494)(359,496)(360,495)
(361,481)(362,482)(363,484)(364,483)(365,485)(366,486)(367,488)(368,487)
(369,510)(370,509)(371,511)(372,512)(373,506)(374,505)(375,507)(376,508)
(377,502)(378,501)(379,503)(380,504)(381,498)(382,497)(383,499)(384,500);;
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*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(512)!(  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)
(  8, 72)(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)
( 16, 80)( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 87)
( 24, 88)( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)
( 32, 96)( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)
( 40,100)( 41,109)( 42,110)( 43,111)( 44,112)( 45,105)( 46,106)( 47,107)
( 48,108)( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)
( 56,116)( 57,125)( 58,126)( 59,127)( 60,128)( 61,121)( 62,122)( 63,123)
( 64,124)(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)
(136,200)(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)
(144,208)(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)
(152,216)(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)
(160,224)(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)
(168,228)(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)
(176,236)(177,245)(178,246)(179,247)(180,248)(181,241)(182,242)(183,243)
(184,244)(185,253)(186,254)(187,255)(188,256)(189,249)(190,250)(191,251)
(192,252)(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)
(264,328)(265,329)(266,330)(267,331)(268,332)(269,333)(270,334)(271,335)
(272,336)(273,337)(274,338)(275,339)(276,340)(277,341)(278,342)(279,343)
(280,344)(281,345)(282,346)(283,347)(284,348)(285,349)(286,350)(287,351)
(288,352)(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)
(296,356)(297,365)(298,366)(299,367)(300,368)(301,361)(302,362)(303,363)
(304,364)(305,373)(306,374)(307,375)(308,376)(309,369)(310,370)(311,371)
(312,372)(313,381)(314,382)(315,383)(316,384)(317,377)(318,378)(319,379)
(320,380)(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)
(392,456)(393,457)(394,458)(395,459)(396,460)(397,461)(398,462)(399,463)
(400,464)(401,465)(402,466)(403,467)(404,468)(405,469)(406,470)(407,471)
(408,472)(409,473)(410,474)(411,475)(412,476)(413,477)(414,478)(415,479)
(416,480)(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)
(424,484)(425,493)(426,494)(427,495)(428,496)(429,489)(430,490)(431,491)
(432,492)(433,501)(434,502)(435,503)(436,504)(437,497)(438,498)(439,499)
(440,500)(441,509)(442,510)(443,511)(444,512)(445,505)(446,506)(447,507)
(448,508);
s1 := Sym(512)!(  1,257)(  2,258)(  3,260)(  4,259)(  5,262)(  6,261)(  7,263)
(  8,264)(  9,267)( 10,268)( 11,265)( 12,266)( 13,272)( 14,271)( 15,270)
( 16,269)( 17,273)( 18,274)( 19,276)( 20,275)( 21,278)( 22,277)( 23,279)
( 24,280)( 25,283)( 26,284)( 27,281)( 28,282)( 29,288)( 30,287)( 31,286)
( 32,285)( 33,297)( 34,298)( 35,300)( 36,299)( 37,302)( 38,301)( 39,303)
( 40,304)( 41,289)( 42,290)( 43,292)( 44,291)( 45,294)( 46,293)( 47,295)
( 48,296)( 49,313)( 50,314)( 51,316)( 52,315)( 53,318)( 54,317)( 55,319)
( 56,320)( 57,305)( 58,306)( 59,308)( 60,307)( 61,310)( 62,309)( 63,311)
( 64,312)( 65,337)( 66,338)( 67,340)( 68,339)( 69,342)( 70,341)( 71,343)
( 72,344)( 73,347)( 74,348)( 75,345)( 76,346)( 77,352)( 78,351)( 79,350)
( 80,349)( 81,321)( 82,322)( 83,324)( 84,323)( 85,326)( 86,325)( 87,327)
( 88,328)( 89,331)( 90,332)( 91,329)( 92,330)( 93,336)( 94,335)( 95,334)
( 96,333)( 97,378)( 98,377)( 99,379)(100,380)(101,381)(102,382)(103,384)
(104,383)(105,370)(106,369)(107,371)(108,372)(109,373)(110,374)(111,376)
(112,375)(113,362)(114,361)(115,363)(116,364)(117,365)(118,366)(119,368)
(120,367)(121,354)(122,353)(123,355)(124,356)(125,357)(126,358)(127,360)
(128,359)(129,417)(130,418)(131,420)(132,419)(133,422)(134,421)(135,423)
(136,424)(137,427)(138,428)(139,425)(140,426)(141,432)(142,431)(143,430)
(144,429)(145,433)(146,434)(147,436)(148,435)(149,438)(150,437)(151,439)
(152,440)(153,443)(154,444)(155,441)(156,442)(157,448)(158,447)(159,446)
(160,445)(161,385)(162,386)(163,388)(164,387)(165,390)(166,389)(167,391)
(168,392)(169,395)(170,396)(171,393)(172,394)(173,400)(174,399)(175,398)
(176,397)(177,401)(178,402)(179,404)(180,403)(181,406)(182,405)(183,407)
(184,408)(185,411)(186,412)(187,409)(188,410)(189,416)(190,415)(191,414)
(192,413)(193,501)(194,502)(195,504)(196,503)(197,498)(198,497)(199,499)
(200,500)(201,511)(202,512)(203,509)(204,510)(205,508)(206,507)(207,506)
(208,505)(209,485)(210,486)(211,488)(212,487)(213,482)(214,481)(215,483)
(216,484)(217,495)(218,496)(219,493)(220,494)(221,492)(222,491)(223,490)
(224,489)(225,470)(226,469)(227,471)(228,472)(229,465)(230,466)(231,468)
(232,467)(233,480)(234,479)(235,478)(236,477)(237,475)(238,476)(239,473)
(240,474)(241,454)(242,453)(243,455)(244,456)(245,449)(246,450)(247,452)
(248,451)(249,464)(250,463)(251,462)(252,461)(253,459)(254,460)(255,457)
(256,458);
s2 := Sym(512)!(  1,129)(  2,130)(  3,132)(  4,131)(  5,133)(  6,134)(  7,136)
(  8,135)(  9,139)( 10,140)( 11,137)( 12,138)( 13,143)( 14,144)( 15,141)
( 16,142)( 17,150)( 18,149)( 19,151)( 20,152)( 21,146)( 22,145)( 23,147)
( 24,148)( 25,160)( 26,159)( 27,158)( 28,157)( 29,156)( 30,155)( 31,154)
( 32,153)( 33,169)( 34,170)( 35,172)( 36,171)( 37,173)( 38,174)( 39,176)
( 40,175)( 41,161)( 42,162)( 43,164)( 44,163)( 45,165)( 46,166)( 47,168)
( 48,167)( 49,190)( 50,189)( 51,191)( 52,192)( 53,186)( 54,185)( 55,187)
( 56,188)( 57,182)( 58,181)( 59,183)( 60,184)( 61,178)( 62,177)( 63,179)
( 64,180)( 65,193)( 66,194)( 67,196)( 68,195)( 69,197)( 70,198)( 71,200)
( 72,199)( 73,203)( 74,204)( 75,201)( 76,202)( 77,207)( 78,208)( 79,205)
( 80,206)( 81,214)( 82,213)( 83,215)( 84,216)( 85,210)( 86,209)( 87,211)
( 88,212)( 89,224)( 90,223)( 91,222)( 92,221)( 93,220)( 94,219)( 95,218)
( 96,217)( 97,233)( 98,234)( 99,236)(100,235)(101,237)(102,238)(103,240)
(104,239)(105,225)(106,226)(107,228)(108,227)(109,229)(110,230)(111,232)
(112,231)(113,254)(114,253)(115,255)(116,256)(117,250)(118,249)(119,251)
(120,252)(121,246)(122,245)(123,247)(124,248)(125,242)(126,241)(127,243)
(128,244)(257,385)(258,386)(259,388)(260,387)(261,389)(262,390)(263,392)
(264,391)(265,395)(266,396)(267,393)(268,394)(269,399)(270,400)(271,397)
(272,398)(273,406)(274,405)(275,407)(276,408)(277,402)(278,401)(279,403)
(280,404)(281,416)(282,415)(283,414)(284,413)(285,412)(286,411)(287,410)
(288,409)(289,425)(290,426)(291,428)(292,427)(293,429)(294,430)(295,432)
(296,431)(297,417)(298,418)(299,420)(300,419)(301,421)(302,422)(303,424)
(304,423)(305,446)(306,445)(307,447)(308,448)(309,442)(310,441)(311,443)
(312,444)(313,438)(314,437)(315,439)(316,440)(317,434)(318,433)(319,435)
(320,436)(321,449)(322,450)(323,452)(324,451)(325,453)(326,454)(327,456)
(328,455)(329,459)(330,460)(331,457)(332,458)(333,463)(334,464)(335,461)
(336,462)(337,470)(338,469)(339,471)(340,472)(341,466)(342,465)(343,467)
(344,468)(345,480)(346,479)(347,478)(348,477)(349,476)(350,475)(351,474)
(352,473)(353,489)(354,490)(355,492)(356,491)(357,493)(358,494)(359,496)
(360,495)(361,481)(362,482)(363,484)(364,483)(365,485)(366,486)(367,488)
(368,487)(369,510)(370,509)(371,511)(372,512)(373,506)(374,505)(375,507)
(376,508)(377,502)(378,501)(379,503)(380,504)(381,498)(382,497)(383,499)
(384,500);
poly := sub<Sym(512)|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*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2 >; 
 
References : None.
to this polytope