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# Polytope of Type {4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*512
Also Known As : {4,4}(8,0), {4,4|8}. if this polytope has another name.
Group : SmallGroup(512,60809)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 64, 128, 64
Order of s0s1s2 : 16
Order of s0s1s2s1 : 8
Special Properties :
Toroidal
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4}*256
4-fold quotients : {4,4}*128
8-fold quotients : {4,4}*64
16-fold quotients : {4,4}*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 := (  1,129)(  2,130)(  3,131)(  4,132)(  5,135)(  6,136)(  7,133)(  8,134)
(  9,137)( 10,138)( 11,139)( 12,140)( 13,143)( 14,144)( 15,141)( 16,142)
( 17,159)( 18,160)( 19,157)( 20,158)( 21,154)( 22,153)( 23,156)( 24,155)
( 25,150)( 26,149)( 27,152)( 28,151)( 29,147)( 30,148)( 31,145)( 32,146)
( 33,161)( 34,162)( 35,163)( 36,164)( 37,167)( 38,168)( 39,165)( 40,166)
( 41,169)( 42,170)( 43,171)( 44,172)( 45,175)( 46,176)( 47,173)( 48,174)
( 49,191)( 50,192)( 51,189)( 52,190)( 53,186)( 54,185)( 55,188)( 56,187)
( 57,182)( 58,181)( 59,184)( 60,183)( 61,179)( 62,180)( 63,177)( 64,178)
( 65,193)( 66,194)( 67,195)( 68,196)( 69,199)( 70,200)( 71,197)( 72,198)
( 73,201)( 74,202)( 75,203)( 76,204)( 77,207)( 78,208)( 79,205)( 80,206)
( 81,223)( 82,224)( 83,221)( 84,222)( 85,218)( 86,217)( 87,220)( 88,219)
( 89,214)( 90,213)( 91,216)( 92,215)( 93,211)( 94,212)( 95,209)( 96,210)
( 97,225)( 98,226)( 99,227)(100,228)(101,231)(102,232)(103,229)(104,230)
(105,233)(106,234)(107,235)(108,236)(109,239)(110,240)(111,237)(112,238)
(113,255)(114,256)(115,253)(116,254)(117,250)(118,249)(119,252)(120,251)
(121,246)(122,245)(123,248)(124,247)(125,243)(126,244)(127,241)(128,242)
(257,385)(258,386)(259,387)(260,388)(261,391)(262,392)(263,389)(264,390)
(265,393)(266,394)(267,395)(268,396)(269,399)(270,400)(271,397)(272,398)
(273,415)(274,416)(275,413)(276,414)(277,410)(278,409)(279,412)(280,411)
(281,406)(282,405)(283,408)(284,407)(285,403)(286,404)(287,401)(288,402)
(289,417)(290,418)(291,419)(292,420)(293,423)(294,424)(295,421)(296,422)
(297,425)(298,426)(299,427)(300,428)(301,431)(302,432)(303,429)(304,430)
(305,447)(306,448)(307,445)(308,446)(309,442)(310,441)(311,444)(312,443)
(313,438)(314,437)(315,440)(316,439)(317,435)(318,436)(319,433)(320,434)
(321,449)(322,450)(323,451)(324,452)(325,455)(326,456)(327,453)(328,454)
(329,457)(330,458)(331,459)(332,460)(333,463)(334,464)(335,461)(336,462)
(337,479)(338,480)(339,477)(340,478)(341,474)(342,473)(343,476)(344,475)
(345,470)(346,469)(347,472)(348,471)(349,467)(350,468)(351,465)(352,466)
(353,481)(354,482)(355,483)(356,484)(357,487)(358,488)(359,485)(360,486)
(361,489)(362,490)(363,491)(364,492)(365,495)(366,496)(367,493)(368,494)
(369,511)(370,512)(371,509)(372,510)(373,506)(374,505)(375,508)(376,507)
(377,502)(378,501)(379,504)(380,503)(381,499)(382,500)(383,497)(384,498);;
s1 := (  1,257)(  2,258)(  3,260)(  4,259)(  5,262)(  6,261)(  7,263)(  8,264)
(  9,269)( 10,270)( 11,272)( 12,271)( 13,265)( 14,266)( 15,268)( 16,267)
( 17,273)( 18,274)( 19,276)( 20,275)( 21,278)( 22,277)( 23,279)( 24,280)
( 25,285)( 26,286)( 27,288)( 28,287)( 29,281)( 30,282)( 31,284)( 32,283)
( 33,289)( 34,290)( 35,292)( 36,291)( 37,294)( 38,293)( 39,295)( 40,296)
( 41,301)( 42,302)( 43,304)( 44,303)( 45,297)( 46,298)( 47,300)( 48,299)
( 49,305)( 50,306)( 51,308)( 52,307)( 53,310)( 54,309)( 55,311)( 56,312)
( 57,317)( 58,318)( 59,320)( 60,319)( 61,313)( 62,314)( 63,316)( 64,315)
( 65,337)( 66,338)( 67,340)( 68,339)( 69,342)( 70,341)( 71,343)( 72,344)
( 73,349)( 74,350)( 75,352)( 76,351)( 77,345)( 78,346)( 79,348)( 80,347)
( 81,321)( 82,322)( 83,324)( 84,323)( 85,326)( 86,325)( 87,327)( 88,328)
( 89,333)( 90,334)( 91,336)( 92,335)( 93,329)( 94,330)( 95,332)( 96,331)
( 97,375)( 98,376)( 99,374)(100,373)(101,371)(102,372)(103,370)(104,369)
(105,380)(106,379)(107,377)(108,378)(109,383)(110,384)(111,382)(112,381)
(113,360)(114,359)(115,357)(116,358)(117,356)(118,355)(119,353)(120,354)
(121,363)(122,364)(123,362)(124,361)(125,368)(126,367)(127,365)(128,366)
(129,417)(130,418)(131,420)(132,419)(133,422)(134,421)(135,423)(136,424)
(137,429)(138,430)(139,432)(140,431)(141,425)(142,426)(143,428)(144,427)
(145,433)(146,434)(147,436)(148,435)(149,438)(150,437)(151,439)(152,440)
(153,445)(154,446)(155,448)(156,447)(157,441)(158,442)(159,444)(160,443)
(161,385)(162,386)(163,388)(164,387)(165,390)(166,389)(167,391)(168,392)
(169,397)(170,398)(171,400)(172,399)(173,393)(174,394)(175,396)(176,395)
(177,401)(178,402)(179,404)(180,403)(181,406)(182,405)(183,407)(184,408)
(185,413)(186,414)(187,416)(188,415)(189,409)(190,410)(191,412)(192,411)
(193,508)(194,507)(195,505)(196,506)(197,511)(198,512)(199,510)(200,509)
(201,501)(202,502)(203,504)(204,503)(205,497)(206,498)(207,500)(208,499)
(209,489)(210,490)(211,492)(212,491)(213,494)(214,493)(215,495)(216,496)
(217,488)(218,487)(219,485)(220,486)(221,484)(222,483)(223,481)(224,482)
(225,479)(226,480)(227,478)(228,477)(229,475)(230,476)(231,474)(232,473)
(233,465)(234,466)(235,468)(236,467)(237,470)(238,469)(239,471)(240,472)
(241,461)(242,462)(243,464)(244,463)(245,457)(246,458)(247,460)(248,459)
(249,451)(250,452)(251,450)(252,449)(253,456)(254,455)(255,453)(256,454);;
s2 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 72)(  6, 71)(  7, 70)(  8, 69)
(  9, 76)( 10, 75)( 11, 74)( 12, 73)( 13, 77)( 14, 78)( 15, 79)( 16, 80)
( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 88)( 22, 87)( 23, 86)( 24, 85)
( 25, 92)( 26, 91)( 27, 90)( 28, 89)( 29, 93)( 30, 94)( 31, 95)( 32, 96)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,112)( 38,111)( 39,110)( 40,109)
( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,104)( 46,103)( 47,102)( 48,101)
( 49,124)( 50,123)( 51,122)( 52,121)( 53,125)( 54,126)( 55,127)( 56,128)
( 57,116)( 58,115)( 59,114)( 60,113)( 61,117)( 62,118)( 63,119)( 64,120)
(129,193)(130,194)(131,195)(132,196)(133,200)(134,199)(135,198)(136,197)
(137,204)(138,203)(139,202)(140,201)(141,205)(142,206)(143,207)(144,208)
(145,209)(146,210)(147,211)(148,212)(149,216)(150,215)(151,214)(152,213)
(153,220)(154,219)(155,218)(156,217)(157,221)(158,222)(159,223)(160,224)
(161,233)(162,234)(163,235)(164,236)(165,240)(166,239)(167,238)(168,237)
(169,225)(170,226)(171,227)(172,228)(173,232)(174,231)(175,230)(176,229)
(177,252)(178,251)(179,250)(180,249)(181,253)(182,254)(183,255)(184,256)
(185,244)(186,243)(187,242)(188,241)(189,245)(190,246)(191,247)(192,248)
(257,321)(258,322)(259,323)(260,324)(261,328)(262,327)(263,326)(264,325)
(265,332)(266,331)(267,330)(268,329)(269,333)(270,334)(271,335)(272,336)
(273,337)(274,338)(275,339)(276,340)(277,344)(278,343)(279,342)(280,341)
(281,348)(282,347)(283,346)(284,345)(285,349)(286,350)(287,351)(288,352)
(289,361)(290,362)(291,363)(292,364)(293,368)(294,367)(295,366)(296,365)
(297,353)(298,354)(299,355)(300,356)(301,360)(302,359)(303,358)(304,357)
(305,380)(306,379)(307,378)(308,377)(309,381)(310,382)(311,383)(312,384)
(313,372)(314,371)(315,370)(316,369)(317,373)(318,374)(319,375)(320,376)
(385,449)(386,450)(387,451)(388,452)(389,456)(390,455)(391,454)(392,453)
(393,460)(394,459)(395,458)(396,457)(397,461)(398,462)(399,463)(400,464)
(401,465)(402,466)(403,467)(404,468)(405,472)(406,471)(407,470)(408,469)
(409,476)(410,475)(411,474)(412,473)(413,477)(414,478)(415,479)(416,480)
(417,489)(418,490)(419,491)(420,492)(421,496)(422,495)(423,494)(424,493)
(425,481)(426,482)(427,483)(428,484)(429,488)(430,487)(431,486)(432,485)
(433,508)(434,507)(435,506)(436,505)(437,509)(438,510)(439,511)(440,512)
(441,500)(442,499)(443,498)(444,497)(445,501)(446,502)(447,503)(448,504);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(512)!(  1,129)(  2,130)(  3,131)(  4,132)(  5,135)(  6,136)(  7,133)
(  8,134)(  9,137)( 10,138)( 11,139)( 12,140)( 13,143)( 14,144)( 15,141)
( 16,142)( 17,159)( 18,160)( 19,157)( 20,158)( 21,154)( 22,153)( 23,156)
( 24,155)( 25,150)( 26,149)( 27,152)( 28,151)( 29,147)( 30,148)( 31,145)
( 32,146)( 33,161)( 34,162)( 35,163)( 36,164)( 37,167)( 38,168)( 39,165)
( 40,166)( 41,169)( 42,170)( 43,171)( 44,172)( 45,175)( 46,176)( 47,173)
( 48,174)( 49,191)( 50,192)( 51,189)( 52,190)( 53,186)( 54,185)( 55,188)
( 56,187)( 57,182)( 58,181)( 59,184)( 60,183)( 61,179)( 62,180)( 63,177)
( 64,178)( 65,193)( 66,194)( 67,195)( 68,196)( 69,199)( 70,200)( 71,197)
( 72,198)( 73,201)( 74,202)( 75,203)( 76,204)( 77,207)( 78,208)( 79,205)
( 80,206)( 81,223)( 82,224)( 83,221)( 84,222)( 85,218)( 86,217)( 87,220)
( 88,219)( 89,214)( 90,213)( 91,216)( 92,215)( 93,211)( 94,212)( 95,209)
( 96,210)( 97,225)( 98,226)( 99,227)(100,228)(101,231)(102,232)(103,229)
(104,230)(105,233)(106,234)(107,235)(108,236)(109,239)(110,240)(111,237)
(112,238)(113,255)(114,256)(115,253)(116,254)(117,250)(118,249)(119,252)
(120,251)(121,246)(122,245)(123,248)(124,247)(125,243)(126,244)(127,241)
(128,242)(257,385)(258,386)(259,387)(260,388)(261,391)(262,392)(263,389)
(264,390)(265,393)(266,394)(267,395)(268,396)(269,399)(270,400)(271,397)
(272,398)(273,415)(274,416)(275,413)(276,414)(277,410)(278,409)(279,412)
(280,411)(281,406)(282,405)(283,408)(284,407)(285,403)(286,404)(287,401)
(288,402)(289,417)(290,418)(291,419)(292,420)(293,423)(294,424)(295,421)
(296,422)(297,425)(298,426)(299,427)(300,428)(301,431)(302,432)(303,429)
(304,430)(305,447)(306,448)(307,445)(308,446)(309,442)(310,441)(311,444)
(312,443)(313,438)(314,437)(315,440)(316,439)(317,435)(318,436)(319,433)
(320,434)(321,449)(322,450)(323,451)(324,452)(325,455)(326,456)(327,453)
(328,454)(329,457)(330,458)(331,459)(332,460)(333,463)(334,464)(335,461)
(336,462)(337,479)(338,480)(339,477)(340,478)(341,474)(342,473)(343,476)
(344,475)(345,470)(346,469)(347,472)(348,471)(349,467)(350,468)(351,465)
(352,466)(353,481)(354,482)(355,483)(356,484)(357,487)(358,488)(359,485)
(360,486)(361,489)(362,490)(363,491)(364,492)(365,495)(366,496)(367,493)
(368,494)(369,511)(370,512)(371,509)(372,510)(373,506)(374,505)(375,508)
(376,507)(377,502)(378,501)(379,504)(380,503)(381,499)(382,500)(383,497)
(384,498);
s1 := Sym(512)!(  1,257)(  2,258)(  3,260)(  4,259)(  5,262)(  6,261)(  7,263)
(  8,264)(  9,269)( 10,270)( 11,272)( 12,271)( 13,265)( 14,266)( 15,268)
( 16,267)( 17,273)( 18,274)( 19,276)( 20,275)( 21,278)( 22,277)( 23,279)
( 24,280)( 25,285)( 26,286)( 27,288)( 28,287)( 29,281)( 30,282)( 31,284)
( 32,283)( 33,289)( 34,290)( 35,292)( 36,291)( 37,294)( 38,293)( 39,295)
( 40,296)( 41,301)( 42,302)( 43,304)( 44,303)( 45,297)( 46,298)( 47,300)
( 48,299)( 49,305)( 50,306)( 51,308)( 52,307)( 53,310)( 54,309)( 55,311)
( 56,312)( 57,317)( 58,318)( 59,320)( 60,319)( 61,313)( 62,314)( 63,316)
( 64,315)( 65,337)( 66,338)( 67,340)( 68,339)( 69,342)( 70,341)( 71,343)
( 72,344)( 73,349)( 74,350)( 75,352)( 76,351)( 77,345)( 78,346)( 79,348)
( 80,347)( 81,321)( 82,322)( 83,324)( 84,323)( 85,326)( 86,325)( 87,327)
( 88,328)( 89,333)( 90,334)( 91,336)( 92,335)( 93,329)( 94,330)( 95,332)
( 96,331)( 97,375)( 98,376)( 99,374)(100,373)(101,371)(102,372)(103,370)
(104,369)(105,380)(106,379)(107,377)(108,378)(109,383)(110,384)(111,382)
(112,381)(113,360)(114,359)(115,357)(116,358)(117,356)(118,355)(119,353)
(120,354)(121,363)(122,364)(123,362)(124,361)(125,368)(126,367)(127,365)
(128,366)(129,417)(130,418)(131,420)(132,419)(133,422)(134,421)(135,423)
(136,424)(137,429)(138,430)(139,432)(140,431)(141,425)(142,426)(143,428)
(144,427)(145,433)(146,434)(147,436)(148,435)(149,438)(150,437)(151,439)
(152,440)(153,445)(154,446)(155,448)(156,447)(157,441)(158,442)(159,444)
(160,443)(161,385)(162,386)(163,388)(164,387)(165,390)(166,389)(167,391)
(168,392)(169,397)(170,398)(171,400)(172,399)(173,393)(174,394)(175,396)
(176,395)(177,401)(178,402)(179,404)(180,403)(181,406)(182,405)(183,407)
(184,408)(185,413)(186,414)(187,416)(188,415)(189,409)(190,410)(191,412)
(192,411)(193,508)(194,507)(195,505)(196,506)(197,511)(198,512)(199,510)
(200,509)(201,501)(202,502)(203,504)(204,503)(205,497)(206,498)(207,500)
(208,499)(209,489)(210,490)(211,492)(212,491)(213,494)(214,493)(215,495)
(216,496)(217,488)(218,487)(219,485)(220,486)(221,484)(222,483)(223,481)
(224,482)(225,479)(226,480)(227,478)(228,477)(229,475)(230,476)(231,474)
(232,473)(233,465)(234,466)(235,468)(236,467)(237,470)(238,469)(239,471)
(240,472)(241,461)(242,462)(243,464)(244,463)(245,457)(246,458)(247,460)
(248,459)(249,451)(250,452)(251,450)(252,449)(253,456)(254,455)(255,453)
(256,454);
s2 := Sym(512)!(  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 72)(  6, 71)(  7, 70)
(  8, 69)(  9, 76)( 10, 75)( 11, 74)( 12, 73)( 13, 77)( 14, 78)( 15, 79)
( 16, 80)( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 88)( 22, 87)( 23, 86)
( 24, 85)( 25, 92)( 26, 91)( 27, 90)( 28, 89)( 29, 93)( 30, 94)( 31, 95)
( 32, 96)( 33,105)( 34,106)( 35,107)( 36,108)( 37,112)( 38,111)( 39,110)
( 40,109)( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,104)( 46,103)( 47,102)
( 48,101)( 49,124)( 50,123)( 51,122)( 52,121)( 53,125)( 54,126)( 55,127)
( 56,128)( 57,116)( 58,115)( 59,114)( 60,113)( 61,117)( 62,118)( 63,119)
( 64,120)(129,193)(130,194)(131,195)(132,196)(133,200)(134,199)(135,198)
(136,197)(137,204)(138,203)(139,202)(140,201)(141,205)(142,206)(143,207)
(144,208)(145,209)(146,210)(147,211)(148,212)(149,216)(150,215)(151,214)
(152,213)(153,220)(154,219)(155,218)(156,217)(157,221)(158,222)(159,223)
(160,224)(161,233)(162,234)(163,235)(164,236)(165,240)(166,239)(167,238)
(168,237)(169,225)(170,226)(171,227)(172,228)(173,232)(174,231)(175,230)
(176,229)(177,252)(178,251)(179,250)(180,249)(181,253)(182,254)(183,255)
(184,256)(185,244)(186,243)(187,242)(188,241)(189,245)(190,246)(191,247)
(192,248)(257,321)(258,322)(259,323)(260,324)(261,328)(262,327)(263,326)
(264,325)(265,332)(266,331)(267,330)(268,329)(269,333)(270,334)(271,335)
(272,336)(273,337)(274,338)(275,339)(276,340)(277,344)(278,343)(279,342)
(280,341)(281,348)(282,347)(283,346)(284,345)(285,349)(286,350)(287,351)
(288,352)(289,361)(290,362)(291,363)(292,364)(293,368)(294,367)(295,366)
(296,365)(297,353)(298,354)(299,355)(300,356)(301,360)(302,359)(303,358)
(304,357)(305,380)(306,379)(307,378)(308,377)(309,381)(310,382)(311,383)
(312,384)(313,372)(314,371)(315,370)(316,369)(317,373)(318,374)(319,375)
(320,376)(385,449)(386,450)(387,451)(388,452)(389,456)(390,455)(391,454)
(392,453)(393,460)(394,459)(395,458)(396,457)(397,461)(398,462)(399,463)
(400,464)(401,465)(402,466)(403,467)(404,468)(405,472)(406,471)(407,470)
(408,469)(409,476)(410,475)(411,474)(412,473)(413,477)(414,478)(415,479)
(416,480)(417,489)(418,490)(419,491)(420,492)(421,496)(422,495)(423,494)
(424,493)(425,481)(426,482)(427,483)(428,484)(429,488)(430,487)(431,486)
(432,485)(433,508)(434,507)(435,506)(436,505)(437,509)(438,510)(439,511)
(440,512)(441,500)(442,499)(443,498)(444,497)(445,501)(446,502)(447,503)
(448,504);
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,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

```
References : None.
to this polytope