Polytope of Type {18,12}

Play with this polytope as a twisty puzzle

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

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