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Polytope of Type {6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*648h
if this polytope has a name.
Group : SmallGroup(648,301)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 18, 162, 54
Order of s0s1s2 : 6
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,18,2} of size 1296
Vertex Figure Of :
   {2,6,18} of size 1296
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,18}*324
   3-fold quotients : {6,6}*216c
   6-fold quotients : {3,6}*108
   9-fold quotients : {6,6}*72c
   18-fold quotients : {3,6}*36
   27-fold quotients : {6,2}*24
   54-fold quotients : {3,2}*12
   81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,18}*1296d, {6,36}*1296h
   3-fold covers : {6,18}*1944b, {18,18}*1944h, {18,18}*1944l, {18,18}*1944y, {18,18}*1944ab, {6,18}*1944k, {6,18}*1944u
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 57)( 29, 56)( 30, 55)( 31, 59)
( 32, 58)( 33, 60)( 34, 61)( 35, 63)( 36, 62)( 37, 75)( 38, 74)( 39, 73)
( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 66)( 47, 65)
( 48, 64)( 49, 68)( 50, 67)( 51, 69)( 52, 70)( 53, 72)( 54, 71)( 83, 84)
( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)
( 97,107)( 98,106)( 99,108)(109,138)(110,137)(111,136)(112,140)(113,139)
(114,141)(115,142)(116,144)(117,143)(118,156)(119,155)(120,154)(121,158)
(122,157)(123,159)(124,160)(125,162)(126,161)(127,147)(128,146)(129,145)
(130,149)(131,148)(132,150)(133,151)(134,153)(135,152);;
s1 := (  1,148)(  2,150)(  3,149)(  4,151)(  5,153)(  6,152)(  7,145)(  8,147)
(  9,146)( 10,142)( 11,144)( 12,143)( 13,136)( 14,138)( 15,137)( 16,139)
( 17,141)( 18,140)( 19,156)( 20,155)( 21,154)( 22,159)( 23,158)( 24,157)
( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,124)( 32,126)
( 33,125)( 34,118)( 35,120)( 36,119)( 37,115)( 38,117)( 39,116)( 40,109)
( 41,111)( 42,110)( 43,112)( 44,114)( 45,113)( 46,129)( 47,128)( 48,127)
( 49,132)( 50,131)( 51,130)( 52,135)( 53,134)( 54,133)( 55, 94)( 56, 96)
( 57, 95)( 58, 97)( 59, 99)( 60, 98)( 61, 91)( 62, 93)( 63, 92)( 64, 88)
( 65, 90)( 66, 89)( 67, 82)( 68, 84)( 69, 83)( 70, 85)( 71, 87)( 72, 86)
( 73,102)( 74,101)( 75,100)( 76,105)( 77,104)( 78,103)( 79,108)( 80,107)
( 81,106);;
s2 := (  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, 57)( 29, 56)( 30, 55)( 31, 63)
( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 66)( 38, 65)( 39, 64)
( 40, 72)( 41, 71)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 75)( 47, 74)
( 48, 73)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,138)(110,137)(111,136)(112,144)(113,143)
(114,142)(115,141)(116,140)(117,139)(118,147)(119,146)(120,145)(121,153)
(122,152)(123,151)(124,150)(125,149)(126,148)(127,156)(128,155)(129,154)
(130,162)(131,161)(132,160)(133,159)(134,158)(135,157);;
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*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(162)!(  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)
( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 57)( 29, 56)( 30, 55)
( 31, 59)( 32, 58)( 33, 60)( 34, 61)( 35, 63)( 36, 62)( 37, 75)( 38, 74)
( 39, 73)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 66)
( 47, 65)( 48, 64)( 49, 68)( 50, 67)( 51, 69)( 52, 70)( 53, 72)( 54, 71)
( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)
( 96,103)( 97,107)( 98,106)( 99,108)(109,138)(110,137)(111,136)(112,140)
(113,139)(114,141)(115,142)(116,144)(117,143)(118,156)(119,155)(120,154)
(121,158)(122,157)(123,159)(124,160)(125,162)(126,161)(127,147)(128,146)
(129,145)(130,149)(131,148)(132,150)(133,151)(134,153)(135,152);
s1 := Sym(162)!(  1,148)(  2,150)(  3,149)(  4,151)(  5,153)(  6,152)(  7,145)
(  8,147)(  9,146)( 10,142)( 11,144)( 12,143)( 13,136)( 14,138)( 15,137)
( 16,139)( 17,141)( 18,140)( 19,156)( 20,155)( 21,154)( 22,159)( 23,158)
( 24,157)( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,124)
( 32,126)( 33,125)( 34,118)( 35,120)( 36,119)( 37,115)( 38,117)( 39,116)
( 40,109)( 41,111)( 42,110)( 43,112)( 44,114)( 45,113)( 46,129)( 47,128)
( 48,127)( 49,132)( 50,131)( 51,130)( 52,135)( 53,134)( 54,133)( 55, 94)
( 56, 96)( 57, 95)( 58, 97)( 59, 99)( 60, 98)( 61, 91)( 62, 93)( 63, 92)
( 64, 88)( 65, 90)( 66, 89)( 67, 82)( 68, 84)( 69, 83)( 70, 85)( 71, 87)
( 72, 86)( 73,102)( 74,101)( 75,100)( 76,105)( 77,104)( 78,103)( 79,108)
( 80,107)( 81,106);
s2 := 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, 57)( 29, 56)( 30, 55)
( 31, 63)( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 66)( 38, 65)
( 39, 64)( 40, 72)( 41, 71)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 75)
( 47, 74)( 48, 73)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,138)(110,137)(111,136)(112,144)
(113,143)(114,142)(115,141)(116,140)(117,139)(118,147)(119,146)(120,145)
(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,156)(128,155)
(129,154)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 >; 
 
References : None.
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