Polytope of Type {12,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*864b
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 24, 216, 36
Order of s0s1s2 : 18
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,18,2} of size 1728
Vertex Figure Of :
   {2,12,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,9}*432
   3-fold quotients : {4,18}*288, {12,6}*288b
   4-fold quotients : {6,18}*216b
   6-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c, {12,3}*144
   8-fold quotients : {6,9}*108
   9-fold quotients : {4,6}*96
   12-fold quotients : {4,9}*72, {2,18}*72, {6,6}*72b
   18-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   24-fold quotients : {2,9}*36, {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 : {12,36}*1728f, {12,36}*1728g, {24,18}*1728b, {24,18}*1728d, {12,18}*1728d
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 2.
      24 facets:
         12 of {12}*24
         12 of {6}*12
      12 vertex figures:
         12 of {18}*36
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      18 facets:
         18 of {12}*24
      12 vertex figures:
         12 of {18}*36
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {12}*24
      12 vertex figures:
         12 of {18}*36

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