Polytope of Type {4,66,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,66,2}*1056a
if this polytope has a name.
Group : SmallGroup(1056,998)
Rank : 4
Schlafli Type : {4,66,2}
Number of vertices, edges, etc : 4, 132, 66, 2
Order of s₀s₁s₂s₃ : 132
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 : {2,66,2}*528
   3-fold quotients : {4,22,2}*352
   4-fold quotients : {2,33,2}*264
   6-fold quotients : {2,22,2}*176
   11-fold quotients : {4,6,2}*96a
   12-fold quotients : {2,11,2}*88
   22-fold quotients : {2,6,2}*48
   33-fold quotients : {4,2,2}*32
   44-fold quotients : {2,3,2}*24
   66-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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