Polytope of Type {4,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*768b
if this polytope has a name.
Group : SmallGroup(768,1088555)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 64, 192, 96
Order of s0s1s2 : 6
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6}*384a
   4-fold quotients : {4,6}*192a
   8-fold quotients : {4,6}*96
   16-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   32-fold quotients : {4,3}*24, {2,6}*24
   64-fold quotients : {2,3}*12
   96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 2.
      52 facets:
         44 of {4}*8
         8 of {2}*4
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      48 facets:
         48 of {4}*8
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 2.
      48 facets:
         48 of {4}*8
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      48 facets:
         48 of {4}*8
      32 vertex figures:
         32 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2> of order 2.
      48 facets:
         48 of {4}*8
      36 vertex figures:
         8 of {3}*6
         28 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      30 facets:
         18 of {4}*8
         12 of {2}*4
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 4.
      24 facets:
         24 of {4}*8
      20 vertex figures:
         12 of {6}*12
         8 of {3}*6
   P/N, where N=<s1*s2*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      20 vertex figures:
         8 of {3}*6
         12 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      26 facets:
         22 of {4}*8
         4 of {2}*4
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      28 facets:
         20 of {4}*8
         8 of {2}*4
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      24 facets:
         24 of {4}*8
      16 vertex figures:
         16 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2> of order 8.
      12 facets:
         12 of {4}*8
      8 vertex figures:
         8 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
      12 facets:
         12 of {4}*8
      8 vertex figures:
         8 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 8.
      12 facets:
         12 of {4}*8
      8 vertex figures:
         8 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
      12 facets:
         12 of {4}*8
      8 vertex figures:
         8 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 8.
      12 facets:
         12 of {4}*8
      12 vertex figures:
         8 of {3}*6
         4 of {6}*12

Permutation Representation (GAP) : 
s0 := (  1,221)(  2,222)(  3,224)(  4,223)(  5,218)(  6,217)(  7,219)(  8,220)(  9,213)( 10,214)( 11,216)( 12,215)( 13,210)( 14,209)( 15,211)( 16,212)( 17,205)( 18,206)( 19,208)( 20,207)( 21,202)( 22,201)( 23,203)( 24,204)( 25,197)( 26,198)( 27,200)( 28,199)( 29,194)( 30,193)( 31,195)( 32,196)( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)( 40,252)( 41,245)( 42,246)( 43,248)( 44,247)( 45,242)( 46,241)( 47,243)( 48,244)( 49,237)( 50,238)( 51,240)( 52,239)( 53,234)( 54,233)( 55,235)( 56,236)( 57,229)( 58,230)( 59,232)( 60,231)( 61,226)( 62,225)( 63,227)( 64,228)( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)( 72,284)( 73,277)( 74,278)( 75,280)( 76,279)( 77,274)( 78,273)( 79,275)( 80,276)( 81,269)( 82,270)( 83,272)( 84,271)( 85,266)( 86,265)( 87,267)( 88,268)( 89,261)( 90,262)( 91,264)( 92,263)( 93,258)( 94,257)( 95,259)( 96,260)( 97,318)( 98,317)( 99,319)(100,320)(101,313)(102,314)(103,316)(104,315)(105,310)(106,309)(107,311)(108,312)(109,305)(110,306)(111,308)(112,307)(113,302)(114,301)(115,303)(116,304)(117,297)(118,298)(119,300)(120,299)(121,294)(122,293)(123,295)(124,296)(125,289)(126,290)(127,292)(128,291)(129,350)(130,349)(131,351)(132,352)(133,345)(134,346)(135,348)(136,347)(137,342)(138,341)(139,343)(140,344)(141,337)(142,338)(143,340)(144,339)(145,334)(146,333)(147,335)(148,336)(149,329)(150,330)(151,332)(152,331)(153,326)(154,325)(155,327)(156,328)(157,321)(158,322)(159,324)(160,323)(161,382)(162,381)(163,383)(164,384)(165,377)(166,378)(167,380)(168,379)(169,374)(170,373)(171,375)(172,376)(173,369)(174,370)(175,372)(176,371)(177,366)(178,365)(179,367)(180,368)(181,361)(182,362)(183,364)(184,363)(185,358)(186,357)(187,359)(188,360)(189,353)(190,354)(191,356)(192,355);;
s1 := (  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)(101,103)(102,104)(109,111)(110,112)(113,121)(114,122)(115,123)(116,124)(117,127)(118,128)(119,125)(120,126)(129,161)(130,162)(131,163)(132,164)(133,167)(134,168)(135,165)(136,166)(137,169)(138,170)(139,171)(140,172)(141,175)(142,176)(143,173)(144,174)(145,185)(146,186)(147,187)(148,188)(149,191)(150,192)(151,189)(152,190)(153,177)(154,178)(155,179)(156,180)(157,183)(158,184)(159,181)(160,182)(197,199)(198,200)(205,207)(206,208)(209,217)(210,218)(211,219)(212,220)(213,223)(214,224)(215,221)(216,222)(225,257)(226,258)(227,259)(228,260)(229,263)(230,264)(231,261)(232,262)(233,265)(234,266)(235,267)(236,268)(237,271)(238,272)(239,269)(240,270)(241,281)(242,282)(243,283)(244,284)(245,287)(246,288)(247,285)(248,286)(249,273)(250,274)(251,275)(252,276)(253,279)(254,280)(255,277)(256,278)(293,295)(294,296)(301,303)(302,304)(305,313)(306,314)(307,315)(308,316)(309,319)(310,320)(311,317)(312,318)(321,353)(322,354)(323,355)(324,356)(325,359)(326,360)(327,357)(328,358)(329,361)(330,362)(331,363)(332,364)(333,367)(334,368)(335,365)(336,366)(337,377)(338,378)(339,379)(340,380)(341,383)(342,384)(343,381)(344,382)(345,369)(346,370)(347,371)(348,372)(349,375)(350,376)(351,373)(352,374);;
s2 := (  1,161)(  2,162)(  3,165)(  4,166)(  5,163)(  6,164)(  7,167)(  8,168)(  9,182)( 10,181)( 11,178)( 12,177)( 13,184)( 14,183)( 15,180)( 16,179)( 17,172)( 18,171)( 19,176)( 20,175)( 21,170)( 22,169)( 23,174)( 24,173)( 25,192)( 26,191)( 27,188)( 28,187)( 29,190)( 30,189)( 31,186)( 32,185)( 33,129)( 34,130)( 35,133)( 36,134)( 37,131)( 38,132)( 39,135)( 40,136)( 41,150)( 42,149)( 43,146)( 44,145)( 45,152)( 46,151)( 47,148)( 48,147)( 49,140)( 50,139)( 51,144)( 52,143)( 53,138)( 54,137)( 55,142)( 56,141)( 57,160)( 58,159)( 59,156)( 60,155)( 61,158)( 62,157)( 63,154)( 64,153)( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)( 70,100)( 71,103)( 72,104)( 73,118)( 74,117)( 75,114)( 76,113)( 77,120)( 78,119)( 79,116)( 80,115)( 81,108)( 82,107)( 83,112)( 84,111)( 85,106)( 86,105)( 87,110)( 88,109)( 89,128)( 90,127)( 91,124)( 92,123)( 93,126)( 94,125)( 95,122)( 96,121)(193,353)(194,354)(195,357)(196,358)(197,355)(198,356)(199,359)(200,360)(201,374)(202,373)(203,370)(204,369)(205,376)(206,375)(207,372)(208,371)(209,364)(210,363)(211,368)(212,367)(213,362)(214,361)(215,366)(216,365)(217,384)(218,383)(219,380)(220,379)(221,382)(222,381)(223,378)(224,377)(225,321)(226,322)(227,325)(228,326)(229,323)(230,324)(231,327)(232,328)(233,342)(234,341)(235,338)(236,337)(237,344)(238,343)(239,340)(240,339)(241,332)(242,331)(243,336)(244,335)(245,330)(246,329)(247,334)(248,333)(249,352)(250,351)(251,348)(252,347)(253,350)(254,349)(255,346)(256,345)(257,289)(258,290)(259,293)(260,294)(261,291)(262,292)(263,295)(264,296)(265,310)(266,309)(267,306)(268,305)(269,312)(270,311)(271,308)(272,307)(273,300)(274,299)(275,304)(276,303)(277,298)(278,297)(279,302)(280,301)(281,320)(282,319)(283,316)(284,315)(285,318)(286,317)(287,314)(288,313);;
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*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(384)!(  1,221)(  2,222)(  3,224)(  4,223)(  5,218)(  6,217)(  7,219)(  8,220)(  9,213)( 10,214)( 11,216)( 12,215)( 13,210)( 14,209)( 15,211)( 16,212)( 17,205)( 18,206)( 19,208)( 20,207)( 21,202)( 22,201)( 23,203)( 24,204)( 25,197)( 26,198)( 27,200)( 28,199)( 29,194)( 30,193)( 31,195)( 32,196)( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)( 40,252)( 41,245)( 42,246)( 43,248)( 44,247)( 45,242)( 46,241)( 47,243)( 48,244)( 49,237)( 50,238)( 51,240)( 52,239)( 53,234)( 54,233)( 55,235)( 56,236)( 57,229)( 58,230)( 59,232)( 60,231)( 61,226)( 62,225)( 63,227)( 64,228)( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)( 72,284)( 73,277)( 74,278)( 75,280)( 76,279)( 77,274)( 78,273)( 79,275)( 80,276)( 81,269)( 82,270)( 83,272)( 84,271)( 85,266)( 86,265)( 87,267)( 88,268)( 89,261)( 90,262)( 91,264)( 92,263)( 93,258)( 94,257)( 95,259)( 96,260)( 97,318)( 98,317)( 99,319)(100,320)(101,313)(102,314)(103,316)(104,315)(105,310)(106,309)(107,311)(108,312)(109,305)(110,306)(111,308)(112,307)(113,302)(114,301)(115,303)(116,304)(117,297)(118,298)(119,300)(120,299)(121,294)(122,293)(123,295)(124,296)(125,289)(126,290)(127,292)(128,291)(129,350)(130,349)(131,351)(132,352)(133,345)(134,346)(135,348)(136,347)(137,342)(138,341)(139,343)(140,344)(141,337)(142,338)(143,340)(144,339)(145,334)(146,333)(147,335)(148,336)(149,329)(150,330)(151,332)(152,331)(153,326)(154,325)(155,327)(156,328)(157,321)(158,322)(159,324)(160,323)(161,382)(162,381)(163,383)(164,384)(165,377)(166,378)(167,380)(168,379)(169,374)(170,373)(171,375)(172,376)(173,369)(174,370)(175,372)(176,371)(177,366)(178,365)(179,367)(180,368)(181,361)(182,362)(183,364)(184,363)(185,358)(186,357)(187,359)(188,360)(189,353)(190,354)(191,356)(192,355);
s1 := Sym(384)!(  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)(101,103)(102,104)(109,111)(110,112)(113,121)(114,122)(115,123)(116,124)(117,127)(118,128)(119,125)(120,126)(129,161)(130,162)(131,163)(132,164)(133,167)(134,168)(135,165)(136,166)(137,169)(138,170)(139,171)(140,172)(141,175)(142,176)(143,173)(144,174)(145,185)(146,186)(147,187)(148,188)(149,191)(150,192)(151,189)(152,190)(153,177)(154,178)(155,179)(156,180)(157,183)(158,184)(159,181)(160,182)(197,199)(198,200)(205,207)(206,208)(209,217)(210,218)(211,219)(212,220)(213,223)(214,224)(215,221)(216,222)(225,257)(226,258)(227,259)(228,260)(229,263)(230,264)(231,261)(232,262)(233,265)(234,266)(235,267)(236,268)(237,271)(238,272)(239,269)(240,270)(241,281)(242,282)(243,283)(244,284)(245,287)(246,288)(247,285)(248,286)(249,273)(250,274)(251,275)(252,276)(253,279)(254,280)(255,277)(256,278)(293,295)(294,296)(301,303)(302,304)(305,313)(306,314)(307,315)(308,316)(309,319)(310,320)(311,317)(312,318)(321,353)(322,354)(323,355)(324,356)(325,359)(326,360)(327,357)(328,358)(329,361)(330,362)(331,363)(332,364)(333,367)(334,368)(335,365)(336,366)(337,377)(338,378)(339,379)(340,380)(341,383)(342,384)(343,381)(344,382)(345,369)(346,370)(347,371)(348,372)(349,375)(350,376)(351,373)(352,374);
s2 := Sym(384)!(  1,161)(  2,162)(  3,165)(  4,166)(  5,163)(  6,164)(  7,167)(  8,168)(  9,182)( 10,181)( 11,178)( 12,177)( 13,184)( 14,183)( 15,180)( 16,179)( 17,172)( 18,171)( 19,176)( 20,175)( 21,170)( 22,169)( 23,174)( 24,173)( 25,192)( 26,191)( 27,188)( 28,187)( 29,190)( 30,189)( 31,186)( 32,185)( 33,129)( 34,130)( 35,133)( 36,134)( 37,131)( 38,132)( 39,135)( 40,136)( 41,150)( 42,149)( 43,146)( 44,145)( 45,152)( 46,151)( 47,148)( 48,147)( 49,140)( 50,139)( 51,144)( 52,143)( 53,138)( 54,137)( 55,142)( 56,141)( 57,160)( 58,159)( 59,156)( 60,155)( 61,158)( 62,157)( 63,154)( 64,153)( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)( 70,100)( 71,103)( 72,104)( 73,118)( 74,117)( 75,114)( 76,113)( 77,120)( 78,119)( 79,116)( 80,115)( 81,108)( 82,107)( 83,112)( 84,111)( 85,106)( 86,105)( 87,110)( 88,109)( 89,128)( 90,127)( 91,124)( 92,123)( 93,126)( 94,125)( 95,122)( 96,121)(193,353)(194,354)(195,357)(196,358)(197,355)(198,356)(199,359)(200,360)(201,374)(202,373)(203,370)(204,369)(205,376)(206,375)(207,372)(208,371)(209,364)(210,363)(211,368)(212,367)(213,362)(214,361)(215,366)(216,365)(217,384)(218,383)(219,380)(220,379)(221,382)(222,381)(223,378)(224,377)(225,321)(226,322)(227,325)(228,326)(229,323)(230,324)(231,327)(232,328)(233,342)(234,341)(235,338)(236,337)(237,344)(238,343)(239,340)(240,339)(241,332)(242,331)(243,336)(244,335)(245,330)(246,329)(247,334)(248,333)(249,352)(250,351)(251,348)(252,347)(253,350)(254,349)(255,346)(256,345)(257,289)(258,290)(259,293)(260,294)(261,291)(262,292)(263,295)(264,296)(265,310)(266,309)(267,306)(268,305)(269,312)(270,311)(271,308)(272,307)(273,300)(274,299)(275,304)(276,303)(277,298)(278,297)(279,302)(280,301)(281,320)(282,319)(283,316)(284,315)(285,318)(286,317)(287,314)(288,313);
poly := sub<Sym(384)|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*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1 >;
References : None.
to this polytope

Twisty Puzzle