Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1296l
if this polytope has a name.
Group : SmallGroup(1296,2985)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 18, 54, 54, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
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 : {6,3,6}*648b
   3-fold quotients : {6,6,2}*432a, {6,6,6}*432f
   6-fold quotients : {6,3,2}*216, {6,3,6}*216
   9-fold quotients : {2,6,6}*144c, {6,6,2}*144b
   18-fold quotients : {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 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161);;
s1 := (  1, 10)(  2, 11)(  3, 12)(  4, 16)(  5, 17)(  6, 18)(  7, 13)(  8, 14)
(  9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 64)( 29, 65)( 30, 66)( 31, 70)
( 32, 71)( 33, 72)( 34, 67)( 35, 68)( 36, 69)( 37, 55)( 38, 56)( 39, 57)
( 40, 61)( 41, 62)( 42, 63)( 43, 58)( 44, 59)( 45, 60)( 46, 73)( 47, 74)
( 48, 75)( 49, 79)( 50, 80)( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 82, 91)
( 83, 92)( 84, 93)( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)
(103,106)(104,107)(105,108)(109,145)(110,146)(111,147)(112,151)(113,152)
(114,153)(115,148)(116,149)(117,150)(118,136)(119,137)(120,138)(121,142)
(122,143)(123,144)(124,139)(125,140)(126,141)(127,154)(128,155)(129,156)
(130,160)(131,161)(132,162)(133,157)(134,158)(135,159);;
s2 := (  1,112)(  2,113)(  3,114)(  4,109)(  5,110)(  6,111)(  7,115)(  8,116)
(  9,117)( 10,132)( 11,130)( 12,131)( 13,129)( 14,127)( 15,128)( 16,135)
( 17,133)( 18,134)( 19,122)( 20,123)( 21,121)( 22,119)( 23,120)( 24,118)
( 25,125)( 26,126)( 27,124)( 28, 85)( 29, 86)( 30, 87)( 31, 82)( 32, 83)
( 33, 84)( 34, 88)( 35, 89)( 36, 90)( 37,105)( 38,103)( 39,104)( 40,102)
( 41,100)( 42,101)( 43,108)( 44,106)( 45,107)( 46, 95)( 47, 96)( 48, 94)
( 49, 92)( 50, 93)( 51, 91)( 52, 98)( 53, 99)( 54, 97)( 55,139)( 56,140)
( 57,141)( 58,136)( 59,137)( 60,138)( 61,142)( 62,143)( 63,144)( 64,159)
( 65,157)( 66,158)( 67,156)( 68,154)( 69,155)( 70,162)( 71,160)( 72,161)
( 73,149)( 74,150)( 75,148)( 76,146)( 77,147)( 78,145)( 79,152)( 80,153)
( 81,151);;
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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope