Polytope of Type {6,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*768b
if this polytope has a name.
Group : SmallGroup(768,1088555)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 96, 192, 64
Order of s0s1s2 : 6
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   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 : {6,4}*384a
   4-fold quotients : {6,4}*192a
   8-fold quotients : {6,4}*96
   16-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   32-fold quotients : {3,4}*24, {6,2}*24
   64-fold quotients : {3,2}*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=<s1*s0*s1*s2*s1*s0*s2*s1> of order 2.
      32 facets:
         32 of {6}*12
      52 vertex figures:
         44 of {4}*8
         8 of {2}*4
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
      32 facets:
         32 of {6}*12
      48 vertex figures:
         48 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2> of order 2.
      32 facets:
         32 of {6}*12
      48 vertex figures:
         48 of {4}*8
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      32 facets:
         32 of {6}*12
      48 vertex figures:
         48 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
      36 facets:
         8 of {3}*6
         28 of {6}*12
      48 vertex figures:
         48 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
      16 facets:
         16 of {6}*12
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
      16 facets:
         16 of {6}*12
      30 vertex figures:
         18 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 4.
      16 facets:
         16 of {6}*12
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s1*s0*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0> of order 4.
      20 facets:
         12 of {6}*12
         8 of {3}*6
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2> of order 4.
      20 facets:
         8 of {3}*6
         12 of {6}*12
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      16 facets:
         16 of {6}*12
      26 vertex figures:
         22 of {4}*8
         4 of {2}*4
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2> of order 4.
      16 facets:
         16 of {6}*12
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 4.
      16 facets:
         16 of {6}*12
      28 vertex figures:
         20 of {4}*8
         8 of {2}*4
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 4.
      16 facets:
         16 of {6}*12
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 4.
      16 facets:
         16 of {6}*12
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      16 facets:
         16 of {6}*12
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 8.
      8 facets:
         8 of {6}*12
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 8.
      8 facets:
         8 of {6}*12
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 8.
      8 facets:
         8 of {6}*12
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 8.
      8 facets:
         8 of {6}*12
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 8.
      12 facets:
         8 of {3}*6
         4 of {6}*12
      12 vertex figures:
         12 of {4}*8

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

Twisty Puzzle