Part of the Atlas of Small Regular Polytopes

Polytope of Type {16,4}

Atlas Canonical Name {16,4}*512c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(512,58346)
Rank
3
Schläfli Type
{16,4}
Vertices, edges, …
64, 128, 16
Order of s0s1s2
16
Order of s0s1s2s1
8
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

16-fold

32-fold

64-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s2*s1)^4*s2> of order 2

8 facets

32 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s2*s1> of order 2

8 facets

32 vertex figures

P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1> of order 2

8 facets

32 vertex figures

P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1, (s1*s0)^3*s2*s1*s0*s1*s2*s1> of order 4

4 facets

16 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle