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Polytope of Type {18,6}

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