Polytope of Type {12,6}

Play with this polytope as a twisty puzzle

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

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