Polytope of Type {2,54,6}

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