Polytope of Type {6,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*864b
if this polytope has a name.
Group : SmallGroup(864,4000)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 72, 216, 72
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,6,2} of size 1728
Vertex Figure Of :
   {2,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,3}*432
   3-fold quotients : {6,6}*288a
   4-fold quotients : {6,6}*216a
   6-fold quotients : {6,3}*144
   8-fold quotients : {6,3}*108
   9-fold quotients : {6,6}*96
   12-fold quotients : {6,6}*72b
   18-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {6,3}*36
   36-fold quotients : {3,3}*24, {2,6}*24
   72-fold quotients : {2,3}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,12}*1728a, {12,6}*1728c, {6,6}*1728a, {6,12}*1728d, {12,6}*1728e
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
      40 facets:
         8 of {3}*6
         32 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 2.
      36 facets:
         36 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 2.
      36 facets:
         36 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
      24 facets:
         24 of {6}*12
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s0*s1*s0*s1> of order 3.
      28 facets:
         6 of {2}*4
         22 of {6}*12
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 4.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1*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*s0*s2*s1*s0*s2*s1> of order 4.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 4.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2> of order 4.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {6}*12
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2> of order 6.
      16 facets:
         8 of {3}*6
         8 of {6}*12
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 6.
      12 facets:
         12 of {6}*12
      12 vertex figures:
         12 of {6}*12

Permutation Representation (GAP) : 
s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 81)( 42, 83)( 43, 82)( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 85)( 50, 87)( 51, 86)( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)( 60, 92)( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)( 71,102)( 72,104)(110,111)(113,117)(114,119)(115,118)(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)(139,142)(140,144)(145,181)(146,183)(147,182)(148,184)(149,189)(150,191)(151,190)(152,192)(153,185)(154,187)(155,186)(156,188)(157,193)(158,195)(159,194)(160,196)(161,201)(162,203)(163,202)(164,204)(165,197)(166,199)(167,198)(168,200)(169,205)(170,207)(171,206)(172,208)(173,213)(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);;
s1 := (  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 45)( 10, 46)( 11, 48)( 12, 47)( 13, 69)( 14, 70)( 15, 72)( 16, 71)( 17, 61)( 18, 62)( 19, 64)( 20, 63)( 21, 65)( 22, 66)( 23, 68)( 24, 67)( 25, 53)( 26, 54)( 27, 56)( 28, 55)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 75, 76)( 79, 80)( 83, 84)( 85,105)( 86,106)( 87,108)( 88,107)( 89, 97)( 90, 98)( 91,100)( 92, 99)( 93,101)( 94,102)( 95,104)( 96,103)(109,145)(110,146)(111,148)(112,147)(113,149)(114,150)(115,152)(116,151)(117,153)(118,154)(119,156)(120,155)(121,177)(122,178)(123,180)(124,179)(125,169)(126,170)(127,172)(128,171)(129,173)(130,174)(131,176)(132,175)(133,161)(134,162)(135,164)(136,163)(137,165)(138,166)(139,168)(140,167)(141,157)(142,158)(143,160)(144,159)(183,184)(187,188)(191,192)(193,213)(194,214)(195,216)(196,215)(197,205)(198,206)(199,208)(200,207)(201,209)(202,210)(203,212)(204,211);;
s2 := (  1,124)(  2,122)(  3,123)(  4,121)(  5,128)(  6,126)(  7,127)(  8,125)(  9,132)( 10,130)( 11,131)( 12,129)( 13,112)( 14,110)( 15,111)( 16,109)( 17,116)( 18,114)( 19,115)( 20,113)( 21,120)( 22,118)( 23,119)( 24,117)( 25,136)( 26,134)( 27,135)( 28,133)( 29,140)( 30,138)( 31,139)( 32,137)( 33,144)( 34,142)( 35,143)( 36,141)( 37,196)( 38,194)( 39,195)( 40,193)( 41,200)( 42,198)( 43,199)( 44,197)( 45,204)( 46,202)( 47,203)( 48,201)( 49,184)( 50,182)( 51,183)( 52,181)( 53,188)( 54,186)( 55,187)( 56,185)( 57,192)( 58,190)( 59,191)( 60,189)( 61,208)( 62,206)( 63,207)( 64,205)( 65,212)( 66,210)( 67,211)( 68,209)( 69,216)( 70,214)( 71,215)( 72,213)( 73,160)( 74,158)( 75,159)( 76,157)( 77,164)( 78,162)( 79,163)( 80,161)( 81,168)( 82,166)( 83,167)( 84,165)( 85,148)( 86,146)( 87,147)( 88,145)( 89,152)( 90,150)( 91,151)( 92,149)( 93,156)( 94,154)( 95,155)( 96,153)( 97,172)( 98,170)( 99,171)(100,169)(101,176)(102,174)(103,175)(104,173)(105,180)(106,178)(107,179)(108,177);;
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*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*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*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(216)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 81)( 42, 83)( 43, 82)( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 85)( 50, 87)( 51, 86)( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)( 60, 92)( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)( 71,102)( 72,104)(110,111)(113,117)(114,119)(115,118)(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)(139,142)(140,144)(145,181)(146,183)(147,182)(148,184)(149,189)(150,191)(151,190)(152,192)(153,185)(154,187)(155,186)(156,188)(157,193)(158,195)(159,194)(160,196)(161,201)(162,203)(163,202)(164,204)(165,197)(166,199)(167,198)(168,200)(169,205)(170,207)(171,206)(172,208)(173,213)(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);
s1 := Sym(216)!(  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 45)( 10, 46)( 11, 48)( 12, 47)( 13, 69)( 14, 70)( 15, 72)( 16, 71)( 17, 61)( 18, 62)( 19, 64)( 20, 63)( 21, 65)( 22, 66)( 23, 68)( 24, 67)( 25, 53)( 26, 54)( 27, 56)( 28, 55)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 75, 76)( 79, 80)( 83, 84)( 85,105)( 86,106)( 87,108)( 88,107)( 89, 97)( 90, 98)( 91,100)( 92, 99)( 93,101)( 94,102)( 95,104)( 96,103)(109,145)(110,146)(111,148)(112,147)(113,149)(114,150)(115,152)(116,151)(117,153)(118,154)(119,156)(120,155)(121,177)(122,178)(123,180)(124,179)(125,169)(126,170)(127,172)(128,171)(129,173)(130,174)(131,176)(132,175)(133,161)(134,162)(135,164)(136,163)(137,165)(138,166)(139,168)(140,167)(141,157)(142,158)(143,160)(144,159)(183,184)(187,188)(191,192)(193,213)(194,214)(195,216)(196,215)(197,205)(198,206)(199,208)(200,207)(201,209)(202,210)(203,212)(204,211);
s2 := Sym(216)!(  1,124)(  2,122)(  3,123)(  4,121)(  5,128)(  6,126)(  7,127)(  8,125)(  9,132)( 10,130)( 11,131)( 12,129)( 13,112)( 14,110)( 15,111)( 16,109)( 17,116)( 18,114)( 19,115)( 20,113)( 21,120)( 22,118)( 23,119)( 24,117)( 25,136)( 26,134)( 27,135)( 28,133)( 29,140)( 30,138)( 31,139)( 32,137)( 33,144)( 34,142)( 35,143)( 36,141)( 37,196)( 38,194)( 39,195)( 40,193)( 41,200)( 42,198)( 43,199)( 44,197)( 45,204)( 46,202)( 47,203)( 48,201)( 49,184)( 50,182)( 51,183)( 52,181)( 53,188)( 54,186)( 55,187)( 56,185)( 57,192)( 58,190)( 59,191)( 60,189)( 61,208)( 62,206)( 63,207)( 64,205)( 65,212)( 66,210)( 67,211)( 68,209)( 69,216)( 70,214)( 71,215)( 72,213)( 73,160)( 74,158)( 75,159)( 76,157)( 77,164)( 78,162)( 79,163)( 80,161)( 81,168)( 82,166)( 83,167)( 84,165)( 85,148)( 86,146)( 87,147)( 88,145)( 89,152)( 90,150)( 91,151)( 92,149)( 93,156)( 94,154)( 95,155)( 96,153)( 97,172)( 98,170)( 99,171)(100,169)(101,176)(102,174)(103,175)(104,173)(105,180)(106,178)(107,179)(108,177);
poly := sub<Sym(216)|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*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*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*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