Polytope of Type {6,6,6}

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

Permutation Representation (Magma) :

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

References : None.
to this polytope