Polytope of Type {18,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*864
if this polytope has a name.
Group : SmallGroup(864,3998)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 72, 216, 24
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {18,6,2} of size 1728
Vertex Figure Of :
   {2,18,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {9,6}*432
   3-fold quotients : {6,6}*288b
   4-fold quotients : {18,6}*216b
   6-fold quotients : {3,6}*144
   8-fold quotients : {9,6}*108
   9-fold quotients : {6,6}*96
   12-fold quotients : {18,2}*72, {6,6}*72c
   18-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {9,2}*36, {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 : {36,6}*1728a, {18,12}*1728a, {18,6}*1728a, {36,6}*1728c, {18,12}*1728b
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      12 facets:
         12 of {18}*36
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 2.
      12 facets:
         12 of {18}*36
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      8 facets:
         8 of {18}*36
      36 vertex figures:
         18 of {6}*12
         18 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      6 facets:
         6 of {18}*36
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1> of order 4.
      6 facets:
         6 of {18}*36
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      6 facets:
         6 of {18}*36
      18 vertex figures:
         18 of {6}*12

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