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}*864a
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 : {3,6}*432
   3-fold quotients : {6,6}*288b
   4-fold quotients : {6,6}*216c
   6-fold quotients : {3,6}*144
   8-fold quotients : {3,6}*108
   9-fold quotients : {6,6}*96
   12-fold quotients : {6,6}*72c
   18-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {3,6}*36
   36-fold quotients : {3,3}*24, {6,2}*24
   72-fold quotients : {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6}*1728a, {6,12}*1728c, {6,6}*1728b, {12,6}*1728d, {6,12}*1728e
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      36 facets:
         36 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      36 facets:
         36 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      36 facets:
         36 of {6}*12
      40 vertex figures:
         32 of {6}*12
         8 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
      24 facets:
         24 of {6}*12
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {6}*12
      28 vertex figures:
         22 of {6}*12
         6 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 4.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 4.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s2*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*s2*s1*s0*s1*s2*s1*s2> of order 4.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 4.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 6.
      12 facets:
         12 of {6}*12
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {6}*12
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2> of order 6.
      12 facets:
         12 of {6}*12
      16 vertex figures:
         8 of {3}*6
         8 of {6}*12

Permutation Representation (GAP) : 
s0 := (  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53,101)( 54,103)( 55,102)( 56,104)( 57,105)( 58,107)( 59,106)( 60,108)( 61, 85)( 62, 87)( 63, 86)( 64, 88)( 65, 89)( 66, 91)( 67, 90)( 68, 92)( 69, 93)( 70, 95)( 71, 94)( 72, 96)(110,111)(114,115)(118,119)(121,133)(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)(129,141)(130,143)(131,142)(132,144)(145,181)(146,183)(147,182)(148,184)(149,185)(150,187)(151,186)(152,188)(153,189)(154,191)(155,190)(156,192)(157,205)(158,207)(159,206)(160,208)(161,209)(162,211)(163,210)(164,212)(165,213)(166,215)(167,214)(168,216)(169,193)(170,195)(171,194)(172,196)(173,197)(174,199)(175,198)(176,200)(177,201)(178,203)(179,202)(180,204);;
s1 := (  1,197)(  2,200)(  3,199)(  4,198)(  5,201)(  6,204)(  7,203)(  8,202)(  9,193)( 10,196)( 11,195)( 12,194)( 13,189)( 14,192)( 15,191)( 16,190)( 17,181)( 18,184)( 19,183)( 20,182)( 21,185)( 22,188)( 23,187)( 24,186)( 25,205)( 26,208)( 27,207)( 28,206)( 29,209)( 30,212)( 31,211)( 32,210)( 33,213)( 34,216)( 35,215)( 36,214)( 37,161)( 38,164)( 39,163)( 40,162)( 41,165)( 42,168)( 43,167)( 44,166)( 45,157)( 46,160)( 47,159)( 48,158)( 49,153)( 50,156)( 51,155)( 52,154)( 53,145)( 54,148)( 55,147)( 56,146)( 57,149)( 58,152)( 59,151)( 60,150)( 61,169)( 62,172)( 63,171)( 64,170)( 65,173)( 66,176)( 67,175)( 68,174)( 69,177)( 70,180)( 71,179)( 72,178)( 73,125)( 74,128)( 75,127)( 76,126)( 77,129)( 78,132)( 79,131)( 80,130)( 81,121)( 82,124)( 83,123)( 84,122)( 85,117)( 86,120)( 87,119)( 88,118)( 89,109)( 90,112)( 91,111)( 92,110)( 93,113)( 94,116)( 95,115)( 96,114)( 97,133)( 98,136)( 99,135)(100,134)(101,137)(102,140)(103,139)(104,138)(105,141)(106,144)(107,143)(108,142);;
s2 := (  1,  4)(  5, 12)(  6, 10)(  7, 11)(  8,  9)( 13, 16)( 17, 24)( 18, 22)( 19, 23)( 20, 21)( 25, 28)( 29, 36)( 30, 34)( 31, 35)( 32, 33)( 37, 76)( 38, 74)( 39, 75)( 40, 73)( 41, 84)( 42, 82)( 43, 83)( 44, 81)( 45, 80)( 46, 78)( 47, 79)( 48, 77)( 49, 88)( 50, 86)( 51, 87)( 52, 85)( 53, 96)( 54, 94)( 55, 95)( 56, 93)( 57, 92)( 58, 90)( 59, 91)( 60, 89)( 61,100)( 62, 98)( 63, 99)( 64, 97)( 65,108)( 66,106)( 67,107)( 68,105)( 69,104)( 70,102)( 71,103)( 72,101)(109,112)(113,120)(114,118)(115,119)(116,117)(121,124)(125,132)(126,130)(127,131)(128,129)(133,136)(137,144)(138,142)(139,143)(140,141)(145,184)(146,182)(147,183)(148,181)(149,192)(150,190)(151,191)(152,189)(153,188)(154,186)(155,187)(156,185)(157,196)(158,194)(159,195)(160,193)(161,204)(162,202)(163,203)(164,201)(165,200)(166,198)(167,199)(168,197)(169,208)(170,206)(171,207)(172,205)(173,216)(174,214)(175,215)(176,213)(177,212)(178,210)(179,211)(180,209);;
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*s0*s1*s2*s1*s0*s1*s0*s1*s2*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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(216)!(  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53,101)( 54,103)( 55,102)( 56,104)( 57,105)( 58,107)( 59,106)( 60,108)( 61, 85)( 62, 87)( 63, 86)( 64, 88)( 65, 89)( 66, 91)( 67, 90)( 68, 92)( 69, 93)( 70, 95)( 71, 94)( 72, 96)(110,111)(114,115)(118,119)(121,133)(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)(129,141)(130,143)(131,142)(132,144)(145,181)(146,183)(147,182)(148,184)(149,185)(150,187)(151,186)(152,188)(153,189)(154,191)(155,190)(156,192)(157,205)(158,207)(159,206)(160,208)(161,209)(162,211)(163,210)(164,212)(165,213)(166,215)(167,214)(168,216)(169,193)(170,195)(171,194)(172,196)(173,197)(174,199)(175,198)(176,200)(177,201)(178,203)(179,202)(180,204);
s1 := Sym(216)!(  1,197)(  2,200)(  3,199)(  4,198)(  5,201)(  6,204)(  7,203)(  8,202)(  9,193)( 10,196)( 11,195)( 12,194)( 13,189)( 14,192)( 15,191)( 16,190)( 17,181)( 18,184)( 19,183)( 20,182)( 21,185)( 22,188)( 23,187)( 24,186)( 25,205)( 26,208)( 27,207)( 28,206)( 29,209)( 30,212)( 31,211)( 32,210)( 33,213)( 34,216)( 35,215)( 36,214)( 37,161)( 38,164)( 39,163)( 40,162)( 41,165)( 42,168)( 43,167)( 44,166)( 45,157)( 46,160)( 47,159)( 48,158)( 49,153)( 50,156)( 51,155)( 52,154)( 53,145)( 54,148)( 55,147)( 56,146)( 57,149)( 58,152)( 59,151)( 60,150)( 61,169)( 62,172)( 63,171)( 64,170)( 65,173)( 66,176)( 67,175)( 68,174)( 69,177)( 70,180)( 71,179)( 72,178)( 73,125)( 74,128)( 75,127)( 76,126)( 77,129)( 78,132)( 79,131)( 80,130)( 81,121)( 82,124)( 83,123)( 84,122)( 85,117)( 86,120)( 87,119)( 88,118)( 89,109)( 90,112)( 91,111)( 92,110)( 93,113)( 94,116)( 95,115)( 96,114)( 97,133)( 98,136)( 99,135)(100,134)(101,137)(102,140)(103,139)(104,138)(105,141)(106,144)(107,143)(108,142);
s2 := Sym(216)!(  1,  4)(  5, 12)(  6, 10)(  7, 11)(  8,  9)( 13, 16)( 17, 24)( 18, 22)( 19, 23)( 20, 21)( 25, 28)( 29, 36)( 30, 34)( 31, 35)( 32, 33)( 37, 76)( 38, 74)( 39, 75)( 40, 73)( 41, 84)( 42, 82)( 43, 83)( 44, 81)( 45, 80)( 46, 78)( 47, 79)( 48, 77)( 49, 88)( 50, 86)( 51, 87)( 52, 85)( 53, 96)( 54, 94)( 55, 95)( 56, 93)( 57, 92)( 58, 90)( 59, 91)( 60, 89)( 61,100)( 62, 98)( 63, 99)( 64, 97)( 65,108)( 66,106)( 67,107)( 68,105)( 69,104)( 70,102)( 71,103)( 72,101)(109,112)(113,120)(114,118)(115,119)(116,117)(121,124)(125,132)(126,130)(127,131)(128,129)(133,136)(137,144)(138,142)(139,143)(140,141)(145,184)(146,182)(147,183)(148,181)(149,192)(150,190)(151,191)(152,189)(153,188)(154,186)(155,187)(156,185)(157,196)(158,194)(159,195)(160,193)(161,204)(162,202)(163,203)(164,201)(165,200)(166,198)(167,199)(168,197)(169,208)(170,206)(171,207)(172,205)(173,216)(174,214)(175,215)(176,213)(177,212)(178,210)(179,211)(180,209);
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*s0*s1*s2*s1*s0*s1*s0*s1*s2*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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle