Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1296c
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 : {3,6,6}*648c
   3-fold quotients : {6,6,2}*432c, {6,6,6}*432g
   6-fold quotients : {3,6,2}*216, {3,6,6}*216b
   9-fold quotients : {6,2,6}*144, {6,6,2}*144c
   18-fold quotients : {3,2,6}*72, {3,6,2}*72, {6,2,3}*72
   27-fold quotients : {2,2,6}*48, {6,2,2}*48
   36-fold quotients : {3,2,3}*36
   54-fold quotients : {2,2,3}*24, {3,2,2}*24
   81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope