Polytope of Type {6,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18,6}*1296d
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 4
Schlafli Type : {6,18,6}
Number of vertices, edges, etc : 6, 54, 54, 6
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,9,6}*648
   3-fold quotients : {2,18,6}*432b, {6,18,2}*432b, {6,6,6}*432f
   6-fold quotients : {2,9,6}*216, {6,9,2}*216, {6,3,6}*216
   9-fold quotients : {2,18,2}*144, {2,6,6}*144c, {6,6,2}*144b
   18-fold quotients : {2,9,2}*72, {2,3,6}*72, {6,3,2}*72
   27-fold quotients : {2,6,2}*48
   54-fold quotients : {2,3,2}*24
   81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)
( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)
( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)
( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)(113,140)
(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)
(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)
(130,157)(131,158)(132,159)(133,160)(134,161)(135,162);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)
(  9, 32)( 10, 47)( 11, 46)( 12, 48)( 13, 53)( 14, 52)( 15, 54)( 16, 50)
( 17, 49)( 18, 51)( 19, 38)( 20, 37)( 21, 39)( 22, 44)( 23, 43)( 24, 45)
( 25, 41)( 26, 40)( 27, 42)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 74)
( 65, 73)( 66, 75)( 67, 80)( 68, 79)( 69, 81)( 70, 77)( 71, 76)( 72, 78)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,128)( 92,127)( 93,129)( 94,134)( 95,133)( 96,135)( 97,131)
( 98,130)( 99,132)(100,119)(101,118)(102,120)(103,125)(104,124)(105,126)
(106,122)(107,121)(108,123)(137,138)(139,142)(140,144)(141,143)(145,155)
(146,154)(147,156)(148,161)(149,160)(150,162)(151,158)(152,157)(153,159);;
s2 := (  1, 94)(  2, 96)(  3, 95)(  4, 91)(  5, 93)(  6, 92)(  7, 97)(  8, 99)
(  9, 98)( 10, 85)( 11, 87)( 12, 86)( 13, 82)( 14, 84)( 15, 83)( 16, 88)
( 17, 90)( 18, 89)( 19,104)( 20,103)( 21,105)( 22,101)( 23,100)( 24,102)
( 25,107)( 26,106)( 27,108)( 28,148)( 29,150)( 30,149)( 31,145)( 32,147)
( 33,146)( 34,151)( 35,153)( 36,152)( 37,139)( 38,141)( 39,140)( 40,136)
( 41,138)( 42,137)( 43,142)( 44,144)( 45,143)( 46,158)( 47,157)( 48,159)
( 49,155)( 50,154)( 51,156)( 52,161)( 53,160)( 54,162)( 55,121)( 56,123)
( 57,122)( 58,118)( 59,120)( 60,119)( 61,124)( 62,126)( 63,125)( 64,112)
( 65,114)( 66,113)( 67,109)( 68,111)( 69,110)( 70,115)( 71,117)( 72,116)
( 73,131)( 74,130)( 75,132)( 76,128)( 77,127)( 78,129)( 79,134)( 80,133)
( 81,135);;
s3 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)
(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(162)!( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)
( 35, 62)( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)
( 43, 70)( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)
( 51, 78)( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)
(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)
(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)
(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162);
s1 := Sym(162)!(  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)
(  8, 33)(  9, 32)( 10, 47)( 11, 46)( 12, 48)( 13, 53)( 14, 52)( 15, 54)
( 16, 50)( 17, 49)( 18, 51)( 19, 38)( 20, 37)( 21, 39)( 22, 44)( 23, 43)
( 24, 45)( 25, 41)( 26, 40)( 27, 42)( 56, 57)( 58, 61)( 59, 63)( 60, 62)
( 64, 74)( 65, 73)( 66, 75)( 67, 80)( 68, 79)( 69, 81)( 70, 77)( 71, 76)
( 72, 78)( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)
( 89,114)( 90,113)( 91,128)( 92,127)( 93,129)( 94,134)( 95,133)( 96,135)
( 97,131)( 98,130)( 99,132)(100,119)(101,118)(102,120)(103,125)(104,124)
(105,126)(106,122)(107,121)(108,123)(137,138)(139,142)(140,144)(141,143)
(145,155)(146,154)(147,156)(148,161)(149,160)(150,162)(151,158)(152,157)
(153,159);
s2 := Sym(162)!(  1, 94)(  2, 96)(  3, 95)(  4, 91)(  5, 93)(  6, 92)(  7, 97)
(  8, 99)(  9, 98)( 10, 85)( 11, 87)( 12, 86)( 13, 82)( 14, 84)( 15, 83)
( 16, 88)( 17, 90)( 18, 89)( 19,104)( 20,103)( 21,105)( 22,101)( 23,100)
( 24,102)( 25,107)( 26,106)( 27,108)( 28,148)( 29,150)( 30,149)( 31,145)
( 32,147)( 33,146)( 34,151)( 35,153)( 36,152)( 37,139)( 38,141)( 39,140)
( 40,136)( 41,138)( 42,137)( 43,142)( 44,144)( 45,143)( 46,158)( 47,157)
( 48,159)( 49,155)( 50,154)( 51,156)( 52,161)( 53,160)( 54,162)( 55,121)
( 56,123)( 57,122)( 58,118)( 59,120)( 60,119)( 61,124)( 62,126)( 63,125)
( 64,112)( 65,114)( 66,113)( 67,109)( 68,111)( 69,110)( 70,115)( 71,117)
( 72,116)( 73,131)( 74,130)( 75,132)( 76,128)( 77,127)( 78,129)( 79,134)
( 80,133)( 81,135);
s3 := Sym(162)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)
(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)
(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162);
poly := sub<Sym(162)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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*s1*s2*s1*s2 >;

References : None.
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