Polytope of Type {2,54,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,54,6}*1296b
if this polytope has a name.
Group : SmallGroup(1296,1859)
Rank : 4
Schlafli Type : {2,54,6}
Number of vertices, edges, etc : 2, 54, 162, 6
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) :
   2-fold quotients : {2,27,6}*648
   3-fold quotients : {2,54,2}*432, {2,18,6}*432b
   6-fold quotients : {2,27,2}*216, {2,9,6}*216
   9-fold quotients : {2,18,2}*144, {2,6,6}*144c
   18-fold quotients : {2,9,2}*72, {2,3,6}*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 := (1,2);;
s1 := (  4,  5)(  6, 11)(  7, 10)(  8,  9)( 12, 21)( 13, 23)( 14, 22)( 15, 29)
( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 30, 65)( 31, 64)( 32, 63)
( 33, 62)( 34, 61)( 35, 60)( 36, 59)( 37, 58)( 38, 57)( 39, 83)( 40, 82)
( 41, 81)( 42, 80)( 43, 79)( 44, 78)( 45, 77)( 46, 76)( 47, 75)( 48, 74)
( 49, 73)( 50, 72)( 51, 71)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)
( 85, 86)( 87, 92)( 88, 91)( 89, 90)( 93,102)( 94,104)( 95,103)( 96,110)
( 97,109)( 98,108)( 99,107)(100,106)(101,105)(111,146)(112,145)(113,144)
(114,143)(115,142)(116,141)(117,140)(118,139)(119,138)(120,164)(121,163)
(122,162)(123,161)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)
(130,154)(131,153)(132,152)(133,151)(134,150)(135,149)(136,148)(137,147);;
s2 := (  3,120)(  4,122)(  5,121)(  6,128)(  7,127)(  8,126)(  9,125)( 10,124)
( 11,123)( 12,111)( 13,113)( 14,112)( 15,119)( 16,118)( 17,117)( 18,116)
( 19,115)( 20,114)( 21,129)( 22,131)( 23,130)( 24,137)( 25,136)( 26,135)
( 27,134)( 28,133)( 29,132)( 30, 93)( 31, 95)( 32, 94)( 33,101)( 34,100)
( 35, 99)( 36, 98)( 37, 97)( 38, 96)( 39, 84)( 40, 86)( 41, 85)( 42, 92)
( 43, 91)( 44, 90)( 45, 89)( 46, 88)( 47, 87)( 48,102)( 49,104)( 50,103)
( 51,110)( 52,109)( 53,108)( 54,107)( 55,106)( 56,105)( 57,155)( 58,154)
( 59,153)( 60,152)( 61,151)( 62,150)( 63,149)( 64,148)( 65,147)( 66,146)
( 67,145)( 68,144)( 69,143)( 70,142)( 71,141)( 72,140)( 73,139)( 74,138)
( 75,164)( 76,163)( 77,162)( 78,161)( 79,160)( 80,159)( 81,158)( 82,157)
( 83,156);;
s3 := ( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)( 18, 27)( 19, 28)
( 20, 29)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)( 44, 53)( 45, 54)
( 46, 55)( 47, 56)( 66, 75)( 67, 76)( 68, 77)( 69, 78)( 70, 79)( 71, 80)
( 72, 81)( 73, 82)( 74, 83)( 93,102)( 94,103)( 95,104)( 96,105)( 97,106)
( 98,107)( 99,108)(100,109)(101,110)(120,129)(121,130)(122,131)(123,132)
(124,133)(125,134)(126,135)(127,136)(128,137)(147,156)(148,157)(149,158)
(150,159)(151,160)(152,161)(153,162)(154,163)(155,164);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 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*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)!(1,2);
s1 := Sym(164)!(  4,  5)(  6, 11)(  7, 10)(  8,  9)( 12, 21)( 13, 23)( 14, 22)
( 15, 29)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 30, 65)( 31, 64)
( 32, 63)( 33, 62)( 34, 61)( 35, 60)( 36, 59)( 37, 58)( 38, 57)( 39, 83)
( 40, 82)( 41, 81)( 42, 80)( 43, 79)( 44, 78)( 45, 77)( 46, 76)( 47, 75)
( 48, 74)( 49, 73)( 50, 72)( 51, 71)( 52, 70)( 53, 69)( 54, 68)( 55, 67)
( 56, 66)( 85, 86)( 87, 92)( 88, 91)( 89, 90)( 93,102)( 94,104)( 95,103)
( 96,110)( 97,109)( 98,108)( 99,107)(100,106)(101,105)(111,146)(112,145)
(113,144)(114,143)(115,142)(116,141)(117,140)(118,139)(119,138)(120,164)
(121,163)(122,162)(123,161)(124,160)(125,159)(126,158)(127,157)(128,156)
(129,155)(130,154)(131,153)(132,152)(133,151)(134,150)(135,149)(136,148)
(137,147);
s2 := Sym(164)!(  3,120)(  4,122)(  5,121)(  6,128)(  7,127)(  8,126)(  9,125)
( 10,124)( 11,123)( 12,111)( 13,113)( 14,112)( 15,119)( 16,118)( 17,117)
( 18,116)( 19,115)( 20,114)( 21,129)( 22,131)( 23,130)( 24,137)( 25,136)
( 26,135)( 27,134)( 28,133)( 29,132)( 30, 93)( 31, 95)( 32, 94)( 33,101)
( 34,100)( 35, 99)( 36, 98)( 37, 97)( 38, 96)( 39, 84)( 40, 86)( 41, 85)
( 42, 92)( 43, 91)( 44, 90)( 45, 89)( 46, 88)( 47, 87)( 48,102)( 49,104)
( 50,103)( 51,110)( 52,109)( 53,108)( 54,107)( 55,106)( 56,105)( 57,155)
( 58,154)( 59,153)( 60,152)( 61,151)( 62,150)( 63,149)( 64,148)( 65,147)
( 66,146)( 67,145)( 68,144)( 69,143)( 70,142)( 71,141)( 72,140)( 73,139)
( 74,138)( 75,164)( 76,163)( 77,162)( 78,161)( 79,160)( 80,159)( 81,158)
( 82,157)( 83,156);
s3 := Sym(164)!( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)( 18, 27)
( 19, 28)( 20, 29)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)( 44, 53)
( 45, 54)( 46, 55)( 47, 56)( 66, 75)( 67, 76)( 68, 77)( 69, 78)( 70, 79)
( 71, 80)( 72, 81)( 73, 82)( 74, 83)( 93,102)( 94,103)( 95,104)( 96,105)
( 97,106)( 98,107)( 99,108)(100,109)(101,110)(120,129)(121,130)(122,131)
(123,132)(124,133)(125,134)(126,135)(127,136)(128,137)(147,156)(148,157)
(149,158)(150,159)(151,160)(152,161)(153,162)(154,163)(155,164);
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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
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*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