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}*648f
if this polytope has a name.
Group : SmallGroup(648,300)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 54, 162, 18
Order of s0s1s2 : 18
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 1296
Vertex Figure Of :
   {2,18,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {18,6}*324b
   3-fold quotients : {6,6}*216b
   6-fold quotients : {6,6}*108
   9-fold quotients : {6,6}*72a
   27-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {2,3}*12, {3,2}*12
   81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,12}*1296b, {36,6}*1296f
   3-fold covers : {18,6}*1944c, {18,18}*1944k, {18,18}*1944m, {18,18}*1944v, {18,6}*1944j, {18,6}*1944s
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      6 facets:
         6 of {18}*36
      18 vertex figures:
         18 of {6}*12

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 56)( 29, 55)( 30, 57)( 31, 62)( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)( 39, 66)( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)( 47, 73)( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,137)(110,136)(111,138)(112,143)(113,142)(114,144)(115,140)(116,139)(117,141)(118,146)(119,145)(120,147)(121,152)(122,151)(123,153)(124,149)(125,148)(126,150)(127,155)(128,154)(129,156)(130,161)(131,160)(132,162)(133,158)(134,157)(135,159);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 33)(  5, 32)(  6, 31)(  7, 35)(  8, 34)(  9, 36)( 10, 53)( 11, 52)( 12, 54)( 13, 46)( 14, 48)( 15, 47)( 16, 51)( 17, 50)( 18, 49)( 19, 40)( 20, 42)( 21, 41)( 22, 45)( 23, 44)( 24, 43)( 25, 38)( 26, 37)( 27, 39)( 55, 56)( 59, 60)( 61, 63)( 64, 81)( 65, 80)( 66, 79)( 67, 74)( 68, 73)( 69, 75)( 70, 76)( 71, 78)( 72, 77)( 82,109)( 83,111)( 84,110)( 85,114)( 86,113)( 87,112)( 88,116)( 89,115)( 90,117)( 91,134)( 92,133)( 93,135)( 94,127)( 95,129)( 96,128)( 97,132)( 98,131)( 99,130)(100,121)(101,123)(102,122)(103,126)(104,125)(105,124)(106,119)(107,118)(108,120)(136,137)(140,141)(142,144)(145,162)(146,161)(147,160)(148,155)(149,154)(150,156)(151,157)(152,159)(153,158);;
s2 := (  1, 91)(  2, 92)(  3, 93)(  4, 97)(  5, 98)(  6, 99)(  7, 94)(  8, 95)(  9, 96)( 10, 82)( 11, 83)( 12, 84)( 13, 88)( 14, 89)( 15, 90)( 16, 85)( 17, 86)( 18, 87)( 19,100)( 20,101)( 21,102)( 22,106)( 23,107)( 24,108)( 25,103)( 26,104)( 27,105)( 28,118)( 29,119)( 30,120)( 31,124)( 32,125)( 33,126)( 34,121)( 35,122)( 36,123)( 37,109)( 38,110)( 39,111)( 40,115)( 41,116)( 42,117)( 43,112)( 44,113)( 45,114)( 46,127)( 47,128)( 48,129)( 49,133)( 50,134)( 51,135)( 52,130)( 53,131)( 54,132)( 55,145)( 56,146)( 57,147)( 58,151)( 59,152)( 60,153)( 61,148)( 62,149)( 63,150)( 64,136)( 65,137)( 66,138)( 67,142)( 68,143)( 69,144)( 70,139)( 71,140)( 72,141)( 73,154)( 74,155)( 75,156)( 76,160)( 77,161)( 78,162)( 79,157)( 80,158)( 81,159);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(162)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 56)( 29, 55)( 30, 57)( 31, 62)( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)( 39, 66)( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)( 47, 73)( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,137)(110,136)(111,138)(112,143)(113,142)(114,144)(115,140)(116,139)(117,141)(118,146)(119,145)(120,147)(121,152)(122,151)(123,153)(124,149)(125,148)(126,150)(127,155)(128,154)(129,156)(130,161)(131,160)(132,162)(133,158)(134,157)(135,159);
s1 := Sym(162)!(  1, 28)(  2, 30)(  3, 29)(  4, 33)(  5, 32)(  6, 31)(  7, 35)(  8, 34)(  9, 36)( 10, 53)( 11, 52)( 12, 54)( 13, 46)( 14, 48)( 15, 47)( 16, 51)( 17, 50)( 18, 49)( 19, 40)( 20, 42)( 21, 41)( 22, 45)( 23, 44)( 24, 43)( 25, 38)( 26, 37)( 27, 39)( 55, 56)( 59, 60)( 61, 63)( 64, 81)( 65, 80)( 66, 79)( 67, 74)( 68, 73)( 69, 75)( 70, 76)( 71, 78)( 72, 77)( 82,109)( 83,111)( 84,110)( 85,114)( 86,113)( 87,112)( 88,116)( 89,115)( 90,117)( 91,134)( 92,133)( 93,135)( 94,127)( 95,129)( 96,128)( 97,132)( 98,131)( 99,130)(100,121)(101,123)(102,122)(103,126)(104,125)(105,124)(106,119)(107,118)(108,120)(136,137)(140,141)(142,144)(145,162)(146,161)(147,160)(148,155)(149,154)(150,156)(151,157)(152,159)(153,158);
s2 := Sym(162)!(  1, 91)(  2, 92)(  3, 93)(  4, 97)(  5, 98)(  6, 99)(  7, 94)(  8, 95)(  9, 96)( 10, 82)( 11, 83)( 12, 84)( 13, 88)( 14, 89)( 15, 90)( 16, 85)( 17, 86)( 18, 87)( 19,100)( 20,101)( 21,102)( 22,106)( 23,107)( 24,108)( 25,103)( 26,104)( 27,105)( 28,118)( 29,119)( 30,120)( 31,124)( 32,125)( 33,126)( 34,121)( 35,122)( 36,123)( 37,109)( 38,110)( 39,111)( 40,115)( 41,116)( 42,117)( 43,112)( 44,113)( 45,114)( 46,127)( 47,128)( 48,129)( 49,133)( 50,134)( 51,135)( 52,130)( 53,131)( 54,132)( 55,145)( 56,146)( 57,147)( 58,151)( 59,152)( 60,153)( 61,148)( 62,149)( 63,150)( 64,136)( 65,137)( 66,138)( 67,142)( 68,143)( 69,144)( 70,139)( 71,140)( 72,141)( 73,154)( 74,155)( 75,156)( 76,160)( 77,161)( 78,162)( 79,157)( 80,158)( 81,159);
poly := sub<Sym(162)|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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1 >;
References : None.
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