Polytope of Type {6,54,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54,2}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1859)
Rank : 4
Schlafli Type : {6,54,2}
Number of vertices, edges, etc : 6, 162, 54, 2
Order of s₀s₁s₂s₃ : 54
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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) :
   3-fold quotients : {2,54,2}*432, {6,18,2}*432a
   6-fold quotients : {2,27,2}*216
   9-fold quotients : {2,18,2}*144, {6,6,2}*144a
   18-fold quotients : {2,9,2}*72
   27-fold quotients : {2,6,2}*48, {6,2,2}*48
   54-fold quotients : {2,3,2}*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)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162);;
s1 := (  1, 10)(  2, 12)(  3, 11)(  4, 18)(  5, 17)(  6, 16)(  7, 15)(  8, 14)
(  9, 13)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 72)( 29, 71)( 30, 70)
( 31, 69)( 32, 68)( 33, 67)( 34, 66)( 35, 65)( 36, 64)( 37, 63)( 38, 62)
( 39, 61)( 40, 60)( 41, 59)( 42, 58)( 43, 57)( 44, 56)( 45, 55)( 46, 81)
( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)
( 82, 91)( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)
( 90, 94)(101,102)(103,108)(104,107)(105,106)(109,153)(110,152)(111,151)
(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)
(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154);;
s2 := (  1,109)(  2,111)(  3,110)(  4,117)(  5,116)(  6,115)(  7,114)(  8,113)
(  9,112)( 10,118)( 11,120)( 12,119)( 13,126)( 14,125)( 15,124)( 16,123)
( 17,122)( 18,121)( 19,127)( 20,129)( 21,128)( 22,135)( 23,134)( 24,133)
( 25,132)( 26,131)( 27,130)( 28, 82)( 29, 84)( 30, 83)( 31, 90)( 32, 89)
( 33, 88)( 34, 87)( 35, 86)( 36, 85)( 37, 91)( 38, 93)( 39, 92)( 40, 99)
( 41, 98)( 42, 97)( 43, 96)( 44, 95)( 45, 94)( 46,100)( 47,102)( 48,101)
( 49,108)( 50,107)( 51,106)( 52,105)( 53,104)( 54,103)( 55,144)( 56,143)
( 57,142)( 58,141)( 59,140)( 60,139)( 61,138)( 62,137)( 63,136)( 64,153)
( 65,152)( 66,151)( 67,150)( 68,149)( 69,148)( 70,147)( 71,146)( 72,145)
( 73,162)( 74,161)( 75,160)( 76,159)( 77,158)( 78,157)( 79,156)( 80,155)
( 81,154);;
s3 := (163,164);;
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*s3*s2*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*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*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*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(164)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162);
s1 := Sym(164)!(  1, 10)(  2, 12)(  3, 11)(  4, 18)(  5, 17)(  6, 16)(  7, 15)
(  8, 14)(  9, 13)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 72)( 29, 71)
( 30, 70)( 31, 69)( 32, 68)( 33, 67)( 34, 66)( 35, 65)( 36, 64)( 37, 63)
( 38, 62)( 39, 61)( 40, 60)( 41, 59)( 42, 58)( 43, 57)( 44, 56)( 45, 55)
( 46, 81)( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)
( 54, 73)( 82, 91)( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)
( 89, 95)( 90, 94)(101,102)(103,108)(104,107)(105,106)(109,153)(110,152)
(111,151)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)
(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)
(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)
(135,154);
s2 := Sym(164)!(  1,109)(  2,111)(  3,110)(  4,117)(  5,116)(  6,115)(  7,114)
(  8,113)(  9,112)( 10,118)( 11,120)( 12,119)( 13,126)( 14,125)( 15,124)
( 16,123)( 17,122)( 18,121)( 19,127)( 20,129)( 21,128)( 22,135)( 23,134)
( 24,133)( 25,132)( 26,131)( 27,130)( 28, 82)( 29, 84)( 30, 83)( 31, 90)
( 32, 89)( 33, 88)( 34, 87)( 35, 86)( 36, 85)( 37, 91)( 38, 93)( 39, 92)
( 40, 99)( 41, 98)( 42, 97)( 43, 96)( 44, 95)( 45, 94)( 46,100)( 47,102)
( 48,101)( 49,108)( 50,107)( 51,106)( 52,105)( 53,104)( 54,103)( 55,144)
( 56,143)( 57,142)( 58,141)( 59,140)( 60,139)( 61,138)( 62,137)( 63,136)
( 64,153)( 65,152)( 66,151)( 67,150)( 68,149)( 69,148)( 70,147)( 71,146)
( 72,145)( 73,162)( 74,161)( 75,160)( 76,159)( 77,158)( 78,157)( 79,156)
( 80,155)( 81,154);
s3 := Sym(164)!(163,164);
poly := sub<Sym(164)|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*s3*s2*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*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope