Polytope of Type {132,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {132,4}*1056c
if this polytope has a name.
Group : SmallGroup(1056,882)
Rank : 3
Schlafli Type : {132,4}
Number of vertices, edges, etc : 132, 264, 4
Order of s0s1s2 : 132
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {66,4}*528b
   4-fold quotients : {33,4}*264
   11-fold quotients : {12,4}*96c
   22-fold quotients : {6,4}*48c
   44-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) : 
s0 := (  3,  4)(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 37)( 10, 38)( 11, 40)( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)( 24, 27)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49,129)( 50,130)( 51,132)( 52,131)( 53,125)( 54,126)( 55,128)( 56,127)( 57,121)( 58,122)( 59,124)( 60,123)( 61,117)( 62,118)( 63,120)( 64,119)( 65,113)( 66,114)( 67,116)( 68,115)( 69,109)( 70,110)( 71,112)( 72,111)( 73,105)( 74,106)( 75,108)( 76,107)( 77,101)( 78,102)( 79,104)( 80,103)( 81, 97)( 82, 98)( 83,100)( 84, 99)( 85, 93)( 86, 94)( 87, 96)( 88, 95)(135,136)(137,173)(138,174)(139,176)(140,175)(141,169)(142,170)(143,172)(144,171)(145,165)(146,166)(147,168)(148,167)(149,161)(150,162)(151,164)(152,163)(153,157)(154,158)(155,160)(156,159)(177,221)(178,222)(179,224)(180,223)(181,261)(182,262)(183,264)(184,263)(185,257)(186,258)(187,260)(188,259)(189,253)(190,254)(191,256)(192,255)(193,249)(194,250)(195,252)(196,251)(197,245)(198,246)(199,248)(200,247)(201,241)(202,242)(203,244)(204,243)(205,237)(206,238)(207,240)(208,239)(209,233)(210,234)(211,236)(212,235)(213,229)(214,230)(215,232)(216,231)(217,225)(218,226)(219,228)(220,227)(265,397)(266,398)(267,400)(268,399)(269,437)(270,438)(271,440)(272,439)(273,433)(274,434)(275,436)(276,435)(277,429)(278,430)(279,432)(280,431)(281,425)(282,426)(283,428)(284,427)(285,421)(286,422)(287,424)(288,423)(289,417)(290,418)(291,420)(292,419)(293,413)(294,414)(295,416)(296,415)(297,409)(298,410)(299,412)(300,411)(301,405)(302,406)(303,408)(304,407)(305,401)(306,402)(307,404)(308,403)(309,485)(310,486)(311,488)(312,487)(313,525)(314,526)(315,528)(316,527)(317,521)(318,522)(319,524)(320,523)(321,517)(322,518)(323,520)(324,519)(325,513)(326,514)(327,516)(328,515)(329,509)(330,510)(331,512)(332,511)(333,505)(334,506)(335,508)(336,507)(337,501)(338,502)(339,504)(340,503)(341,497)(342,498)(343,500)(344,499)(345,493)(346,494)(347,496)(348,495)(349,489)(350,490)(351,492)(352,491)(353,441)(354,442)(355,444)(356,443)(357,481)(358,482)(359,484)(360,483)(361,477)(362,478)(363,480)(364,479)(365,473)(366,474)(367,476)(368,475)(369,469)(370,470)(371,472)(372,471)(373,465)(374,466)(375,468)(376,467)(377,461)(378,462)(379,464)(380,463)(381,457)(382,458)(383,460)(384,459)(385,453)(386,454)(387,456)(388,455)(389,449)(390,450)(391,452)(392,451)(393,445)(394,446)(395,448)(396,447);;
s1 := (  1,313)(  2,316)(  3,315)(  4,314)(  5,309)(  6,312)(  7,311)(  8,310)(  9,349)( 10,352)( 11,351)( 12,350)( 13,345)( 14,348)( 15,347)( 16,346)( 17,341)( 18,344)( 19,343)( 20,342)( 21,337)( 22,340)( 23,339)( 24,338)( 25,333)( 26,336)( 27,335)( 28,334)( 29,329)( 30,332)( 31,331)( 32,330)( 33,325)( 34,328)( 35,327)( 36,326)( 37,321)( 38,324)( 39,323)( 40,322)( 41,317)( 42,320)( 43,319)( 44,318)( 45,269)( 46,272)( 47,271)( 48,270)( 49,265)( 50,268)( 51,267)( 52,266)( 53,305)( 54,308)( 55,307)( 56,306)( 57,301)( 58,304)( 59,303)( 60,302)( 61,297)( 62,300)( 63,299)( 64,298)( 65,293)( 66,296)( 67,295)( 68,294)( 69,289)( 70,292)( 71,291)( 72,290)( 73,285)( 74,288)( 75,287)( 76,286)( 77,281)( 78,284)( 79,283)( 80,282)( 81,277)( 82,280)( 83,279)( 84,278)( 85,273)( 86,276)( 87,275)( 88,274)( 89,357)( 90,360)( 91,359)( 92,358)( 93,353)( 94,356)( 95,355)( 96,354)( 97,393)( 98,396)( 99,395)(100,394)(101,389)(102,392)(103,391)(104,390)(105,385)(106,388)(107,387)(108,386)(109,381)(110,384)(111,383)(112,382)(113,377)(114,380)(115,379)(116,378)(117,373)(118,376)(119,375)(120,374)(121,369)(122,372)(123,371)(124,370)(125,365)(126,368)(127,367)(128,366)(129,361)(130,364)(131,363)(132,362)(133,445)(134,448)(135,447)(136,446)(137,441)(138,444)(139,443)(140,442)(141,481)(142,484)(143,483)(144,482)(145,477)(146,480)(147,479)(148,478)(149,473)(150,476)(151,475)(152,474)(153,469)(154,472)(155,471)(156,470)(157,465)(158,468)(159,467)(160,466)(161,461)(162,464)(163,463)(164,462)(165,457)(166,460)(167,459)(168,458)(169,453)(170,456)(171,455)(172,454)(173,449)(174,452)(175,451)(176,450)(177,401)(178,404)(179,403)(180,402)(181,397)(182,400)(183,399)(184,398)(185,437)(186,440)(187,439)(188,438)(189,433)(190,436)(191,435)(192,434)(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)(200,426)(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)(208,418)(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)(216,410)(217,405)(218,408)(219,407)(220,406)(221,489)(222,492)(223,491)(224,490)(225,485)(226,488)(227,487)(228,486)(229,525)(230,528)(231,527)(232,526)(233,521)(234,524)(235,523)(236,522)(237,517)(238,520)(239,519)(240,518)(241,513)(242,516)(243,515)(244,514)(245,509)(246,512)(247,511)(248,510)(249,505)(250,508)(251,507)(252,506)(253,501)(254,504)(255,503)(256,502)(257,497)(258,500)(259,499)(260,498)(261,493)(262,496)(263,495)(264,494);;
s2 := (  1,134)(  2,133)(  3,136)(  4,135)(  5,138)(  6,137)(  7,140)(  8,139)(  9,142)( 10,141)( 11,144)( 12,143)( 13,146)( 14,145)( 15,148)( 16,147)( 17,150)( 18,149)( 19,152)( 20,151)( 21,154)( 22,153)( 23,156)( 24,155)( 25,158)( 26,157)( 27,160)( 28,159)( 29,162)( 30,161)( 31,164)( 32,163)( 33,166)( 34,165)( 35,168)( 36,167)( 37,170)( 38,169)( 39,172)( 40,171)( 41,174)( 42,173)( 43,176)( 44,175)( 45,178)( 46,177)( 47,180)( 48,179)( 49,182)( 50,181)( 51,184)( 52,183)( 53,186)( 54,185)( 55,188)( 56,187)( 57,190)( 58,189)( 59,192)( 60,191)( 61,194)( 62,193)( 63,196)( 64,195)( 65,198)( 66,197)( 67,200)( 68,199)( 69,202)( 70,201)( 71,204)( 72,203)( 73,206)( 74,205)( 75,208)( 76,207)( 77,210)( 78,209)( 79,212)( 80,211)( 81,214)( 82,213)( 83,216)( 84,215)( 85,218)( 86,217)( 87,220)( 88,219)( 89,222)( 90,221)( 91,224)( 92,223)( 93,226)( 94,225)( 95,228)( 96,227)( 97,230)( 98,229)( 99,232)(100,231)(101,234)(102,233)(103,236)(104,235)(105,238)(106,237)(107,240)(108,239)(109,242)(110,241)(111,244)(112,243)(113,246)(114,245)(115,248)(116,247)(117,250)(118,249)(119,252)(120,251)(121,254)(122,253)(123,256)(124,255)(125,258)(126,257)(127,260)(128,259)(129,262)(130,261)(131,264)(132,263)(265,398)(266,397)(267,400)(268,399)(269,402)(270,401)(271,404)(272,403)(273,406)(274,405)(275,408)(276,407)(277,410)(278,409)(279,412)(280,411)(281,414)(282,413)(283,416)(284,415)(285,418)(286,417)(287,420)(288,419)(289,422)(290,421)(291,424)(292,423)(293,426)(294,425)(295,428)(296,427)(297,430)(298,429)(299,432)(300,431)(301,434)(302,433)(303,436)(304,435)(305,438)(306,437)(307,440)(308,439)(309,442)(310,441)(311,444)(312,443)(313,446)(314,445)(315,448)(316,447)(317,450)(318,449)(319,452)(320,451)(321,454)(322,453)(323,456)(324,455)(325,458)(326,457)(327,460)(328,459)(329,462)(330,461)(331,464)(332,463)(333,466)(334,465)(335,468)(336,467)(337,470)(338,469)(339,472)(340,471)(341,474)(342,473)(343,476)(344,475)(345,478)(346,477)(347,480)(348,479)(349,482)(350,481)(351,484)(352,483)(353,486)(354,485)(355,488)(356,487)(357,490)(358,489)(359,492)(360,491)(361,494)(362,493)(363,496)(364,495)(365,498)(366,497)(367,500)(368,499)(369,502)(370,501)(371,504)(372,503)(373,506)(374,505)(375,508)(376,507)(377,510)(378,509)(379,512)(380,511)(381,514)(382,513)(383,516)(384,515)(385,518)(386,517)(387,520)(388,519)(389,522)(390,521)(391,524)(392,523)(393,526)(394,525)(395,528)(396,527);;
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*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(528)!(  3,  4)(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 37)( 10, 38)( 11, 40)( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)( 24, 27)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49,129)( 50,130)( 51,132)( 52,131)( 53,125)( 54,126)( 55,128)( 56,127)( 57,121)( 58,122)( 59,124)( 60,123)( 61,117)( 62,118)( 63,120)( 64,119)( 65,113)( 66,114)( 67,116)( 68,115)( 69,109)( 70,110)( 71,112)( 72,111)( 73,105)( 74,106)( 75,108)( 76,107)( 77,101)( 78,102)( 79,104)( 80,103)( 81, 97)( 82, 98)( 83,100)( 84, 99)( 85, 93)( 86, 94)( 87, 96)( 88, 95)(135,136)(137,173)(138,174)(139,176)(140,175)(141,169)(142,170)(143,172)(144,171)(145,165)(146,166)(147,168)(148,167)(149,161)(150,162)(151,164)(152,163)(153,157)(154,158)(155,160)(156,159)(177,221)(178,222)(179,224)(180,223)(181,261)(182,262)(183,264)(184,263)(185,257)(186,258)(187,260)(188,259)(189,253)(190,254)(191,256)(192,255)(193,249)(194,250)(195,252)(196,251)(197,245)(198,246)(199,248)(200,247)(201,241)(202,242)(203,244)(204,243)(205,237)(206,238)(207,240)(208,239)(209,233)(210,234)(211,236)(212,235)(213,229)(214,230)(215,232)(216,231)(217,225)(218,226)(219,228)(220,227)(265,397)(266,398)(267,400)(268,399)(269,437)(270,438)(271,440)(272,439)(273,433)(274,434)(275,436)(276,435)(277,429)(278,430)(279,432)(280,431)(281,425)(282,426)(283,428)(284,427)(285,421)(286,422)(287,424)(288,423)(289,417)(290,418)(291,420)(292,419)(293,413)(294,414)(295,416)(296,415)(297,409)(298,410)(299,412)(300,411)(301,405)(302,406)(303,408)(304,407)(305,401)(306,402)(307,404)(308,403)(309,485)(310,486)(311,488)(312,487)(313,525)(314,526)(315,528)(316,527)(317,521)(318,522)(319,524)(320,523)(321,517)(322,518)(323,520)(324,519)(325,513)(326,514)(327,516)(328,515)(329,509)(330,510)(331,512)(332,511)(333,505)(334,506)(335,508)(336,507)(337,501)(338,502)(339,504)(340,503)(341,497)(342,498)(343,500)(344,499)(345,493)(346,494)(347,496)(348,495)(349,489)(350,490)(351,492)(352,491)(353,441)(354,442)(355,444)(356,443)(357,481)(358,482)(359,484)(360,483)(361,477)(362,478)(363,480)(364,479)(365,473)(366,474)(367,476)(368,475)(369,469)(370,470)(371,472)(372,471)(373,465)(374,466)(375,468)(376,467)(377,461)(378,462)(379,464)(380,463)(381,457)(382,458)(383,460)(384,459)(385,453)(386,454)(387,456)(388,455)(389,449)(390,450)(391,452)(392,451)(393,445)(394,446)(395,448)(396,447);
s1 := Sym(528)!(  1,313)(  2,316)(  3,315)(  4,314)(  5,309)(  6,312)(  7,311)(  8,310)(  9,349)( 10,352)( 11,351)( 12,350)( 13,345)( 14,348)( 15,347)( 16,346)( 17,341)( 18,344)( 19,343)( 20,342)( 21,337)( 22,340)( 23,339)( 24,338)( 25,333)( 26,336)( 27,335)( 28,334)( 29,329)( 30,332)( 31,331)( 32,330)( 33,325)( 34,328)( 35,327)( 36,326)( 37,321)( 38,324)( 39,323)( 40,322)( 41,317)( 42,320)( 43,319)( 44,318)( 45,269)( 46,272)( 47,271)( 48,270)( 49,265)( 50,268)( 51,267)( 52,266)( 53,305)( 54,308)( 55,307)( 56,306)( 57,301)( 58,304)( 59,303)( 60,302)( 61,297)( 62,300)( 63,299)( 64,298)( 65,293)( 66,296)( 67,295)( 68,294)( 69,289)( 70,292)( 71,291)( 72,290)( 73,285)( 74,288)( 75,287)( 76,286)( 77,281)( 78,284)( 79,283)( 80,282)( 81,277)( 82,280)( 83,279)( 84,278)( 85,273)( 86,276)( 87,275)( 88,274)( 89,357)( 90,360)( 91,359)( 92,358)( 93,353)( 94,356)( 95,355)( 96,354)( 97,393)( 98,396)( 99,395)(100,394)(101,389)(102,392)(103,391)(104,390)(105,385)(106,388)(107,387)(108,386)(109,381)(110,384)(111,383)(112,382)(113,377)(114,380)(115,379)(116,378)(117,373)(118,376)(119,375)(120,374)(121,369)(122,372)(123,371)(124,370)(125,365)(126,368)(127,367)(128,366)(129,361)(130,364)(131,363)(132,362)(133,445)(134,448)(135,447)(136,446)(137,441)(138,444)(139,443)(140,442)(141,481)(142,484)(143,483)(144,482)(145,477)(146,480)(147,479)(148,478)(149,473)(150,476)(151,475)(152,474)(153,469)(154,472)(155,471)(156,470)(157,465)(158,468)(159,467)(160,466)(161,461)(162,464)(163,463)(164,462)(165,457)(166,460)(167,459)(168,458)(169,453)(170,456)(171,455)(172,454)(173,449)(174,452)(175,451)(176,450)(177,401)(178,404)(179,403)(180,402)(181,397)(182,400)(183,399)(184,398)(185,437)(186,440)(187,439)(188,438)(189,433)(190,436)(191,435)(192,434)(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)(200,426)(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)(208,418)(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)(216,410)(217,405)(218,408)(219,407)(220,406)(221,489)(222,492)(223,491)(224,490)(225,485)(226,488)(227,487)(228,486)(229,525)(230,528)(231,527)(232,526)(233,521)(234,524)(235,523)(236,522)(237,517)(238,520)(239,519)(240,518)(241,513)(242,516)(243,515)(244,514)(245,509)(246,512)(247,511)(248,510)(249,505)(250,508)(251,507)(252,506)(253,501)(254,504)(255,503)(256,502)(257,497)(258,500)(259,499)(260,498)(261,493)(262,496)(263,495)(264,494);
s2 := Sym(528)!(  1,134)(  2,133)(  3,136)(  4,135)(  5,138)(  6,137)(  7,140)(  8,139)(  9,142)( 10,141)( 11,144)( 12,143)( 13,146)( 14,145)( 15,148)( 16,147)( 17,150)( 18,149)( 19,152)( 20,151)( 21,154)( 22,153)( 23,156)( 24,155)( 25,158)( 26,157)( 27,160)( 28,159)( 29,162)( 30,161)( 31,164)( 32,163)( 33,166)( 34,165)( 35,168)( 36,167)( 37,170)( 38,169)( 39,172)( 40,171)( 41,174)( 42,173)( 43,176)( 44,175)( 45,178)( 46,177)( 47,180)( 48,179)( 49,182)( 50,181)( 51,184)( 52,183)( 53,186)( 54,185)( 55,188)( 56,187)( 57,190)( 58,189)( 59,192)( 60,191)( 61,194)( 62,193)( 63,196)( 64,195)( 65,198)( 66,197)( 67,200)( 68,199)( 69,202)( 70,201)( 71,204)( 72,203)( 73,206)( 74,205)( 75,208)( 76,207)( 77,210)( 78,209)( 79,212)( 80,211)( 81,214)( 82,213)( 83,216)( 84,215)( 85,218)( 86,217)( 87,220)( 88,219)( 89,222)( 90,221)( 91,224)( 92,223)( 93,226)( 94,225)( 95,228)( 96,227)( 97,230)( 98,229)( 99,232)(100,231)(101,234)(102,233)(103,236)(104,235)(105,238)(106,237)(107,240)(108,239)(109,242)(110,241)(111,244)(112,243)(113,246)(114,245)(115,248)(116,247)(117,250)(118,249)(119,252)(120,251)(121,254)(122,253)(123,256)(124,255)(125,258)(126,257)(127,260)(128,259)(129,262)(130,261)(131,264)(132,263)(265,398)(266,397)(267,400)(268,399)(269,402)(270,401)(271,404)(272,403)(273,406)(274,405)(275,408)(276,407)(277,410)(278,409)(279,412)(280,411)(281,414)(282,413)(283,416)(284,415)(285,418)(286,417)(287,420)(288,419)(289,422)(290,421)(291,424)(292,423)(293,426)(294,425)(295,428)(296,427)(297,430)(298,429)(299,432)(300,431)(301,434)(302,433)(303,436)(304,435)(305,438)(306,437)(307,440)(308,439)(309,442)(310,441)(311,444)(312,443)(313,446)(314,445)(315,448)(316,447)(317,450)(318,449)(319,452)(320,451)(321,454)(322,453)(323,456)(324,455)(325,458)(326,457)(327,460)(328,459)(329,462)(330,461)(331,464)(332,463)(333,466)(334,465)(335,468)(336,467)(337,470)(338,469)(339,472)(340,471)(341,474)(342,473)(343,476)(344,475)(345,478)(346,477)(347,480)(348,479)(349,482)(350,481)(351,484)(352,483)(353,486)(354,485)(355,488)(356,487)(357,490)(358,489)(359,492)(360,491)(361,494)(362,493)(363,496)(364,495)(365,498)(366,497)(367,500)(368,499)(369,502)(370,501)(371,504)(372,503)(373,506)(374,505)(375,508)(376,507)(377,510)(378,509)(379,512)(380,511)(381,514)(382,513)(383,516)(384,515)(385,518)(386,517)(387,520)(388,519)(389,522)(390,521)(391,524)(392,523)(393,526)(394,525)(395,528)(396,527);
poly := sub<Sym(528)|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*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle