Polytope of Type {2,66,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,66,4}*1056a
if this polytope has a name.
Group : SmallGroup(1056,998)
Rank : 4
Schlafli Type : {2,66,4}
Number of vertices, edges, etc : 2, 66, 132, 4
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 : {2,22,4}*352
   4-fold quotients : {2,33,2}*264
   6-fold quotients : {2,22,2}*176
   11-fold quotients : {2,6,4}*96a
   12-fold quotients : {2,11,2}*88
   22-fold quotients : {2,6,2}*48
   33-fold quotients : {2,2,4}*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 := (1,2);;
s1 := (  4, 13)(  5, 12)(  6, 11)(  7, 10)(  8,  9)( 14, 25)( 15, 35)( 16, 34)
( 17, 33)( 18, 32)( 19, 31)( 20, 30)( 21, 29)( 22, 28)( 23, 27)( 24, 26)
( 37, 46)( 38, 45)( 39, 44)( 40, 43)( 41, 42)( 47, 58)( 48, 68)( 49, 67)
( 50, 66)( 51, 65)( 52, 64)( 53, 63)( 54, 62)( 55, 61)( 56, 60)( 57, 59)
( 70, 79)( 71, 78)( 72, 77)( 73, 76)( 74, 75)( 80, 91)( 81,101)( 82,100)
( 83, 99)( 84, 98)( 85, 97)( 86, 96)( 87, 95)( 88, 94)( 89, 93)( 90, 92)
(103,112)(104,111)(105,110)(106,109)(107,108)(113,124)(114,134)(115,133)
(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125);;
s2 := (  3, 15)(  4, 14)(  5, 24)(  6, 23)(  7, 22)(  8, 21)(  9, 20)( 10, 19)
( 11, 18)( 12, 17)( 13, 16)( 25, 26)( 27, 35)( 28, 34)( 29, 33)( 30, 32)
( 36, 48)( 37, 47)( 38, 57)( 39, 56)( 40, 55)( 41, 54)( 42, 53)( 43, 52)
( 44, 51)( 45, 50)( 46, 49)( 58, 59)( 60, 68)( 61, 67)( 62, 66)( 63, 65)
( 69,114)( 70,113)( 71,123)( 72,122)( 73,121)( 74,120)( 75,119)( 76,118)
( 77,117)( 78,116)( 79,115)( 80,103)( 81,102)( 82,112)( 83,111)( 84,110)
( 85,109)( 86,108)( 87,107)( 88,106)( 89,105)( 90,104)( 91,125)( 92,124)
( 93,134)( 94,133)( 95,132)( 96,131)( 97,130)( 98,129)( 99,128)(100,127)
(101,126);;
s3 := (  3, 69)(  4, 70)(  5, 71)(  6, 72)(  7, 73)(  8, 74)(  9, 75)( 10, 76)
( 11, 77)( 12, 78)( 13, 79)( 14, 80)( 15, 81)( 16, 82)( 17, 83)( 18, 84)
( 19, 85)( 20, 86)( 21, 87)( 22, 88)( 23, 89)( 24, 90)( 25, 91)( 26, 92)
( 27, 93)( 28, 94)( 29, 95)( 30, 96)( 31, 97)( 32, 98)( 33, 99)( 34,100)
( 35,101)( 36,102)( 37,103)( 38,104)( 39,105)( 40,106)( 41,107)( 42,108)
( 43,109)( 44,110)( 45,111)( 46,112)( 47,113)( 48,114)( 49,115)( 50,116)
( 51,117)( 52,118)( 53,119)( 54,120)( 55,121)( 56,122)( 57,123)( 58,124)
( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)( 64,130)( 65,131)( 66,132)
( 67,133)( 68,134);;
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*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(1,2);
s1 := Sym(134)!(  4, 13)(  5, 12)(  6, 11)(  7, 10)(  8,  9)( 14, 25)( 15, 35)
( 16, 34)( 17, 33)( 18, 32)( 19, 31)( 20, 30)( 21, 29)( 22, 28)( 23, 27)
( 24, 26)( 37, 46)( 38, 45)( 39, 44)( 40, 43)( 41, 42)( 47, 58)( 48, 68)
( 49, 67)( 50, 66)( 51, 65)( 52, 64)( 53, 63)( 54, 62)( 55, 61)( 56, 60)
( 57, 59)( 70, 79)( 71, 78)( 72, 77)( 73, 76)( 74, 75)( 80, 91)( 81,101)
( 82,100)( 83, 99)( 84, 98)( 85, 97)( 86, 96)( 87, 95)( 88, 94)( 89, 93)
( 90, 92)(103,112)(104,111)(105,110)(106,109)(107,108)(113,124)(114,134)
(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)
(123,125);
s2 := Sym(134)!(  3, 15)(  4, 14)(  5, 24)(  6, 23)(  7, 22)(  8, 21)(  9, 20)
( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 25, 26)( 27, 35)( 28, 34)( 29, 33)
( 30, 32)( 36, 48)( 37, 47)( 38, 57)( 39, 56)( 40, 55)( 41, 54)( 42, 53)
( 43, 52)( 44, 51)( 45, 50)( 46, 49)( 58, 59)( 60, 68)( 61, 67)( 62, 66)
( 63, 65)( 69,114)( 70,113)( 71,123)( 72,122)( 73,121)( 74,120)( 75,119)
( 76,118)( 77,117)( 78,116)( 79,115)( 80,103)( 81,102)( 82,112)( 83,111)
( 84,110)( 85,109)( 86,108)( 87,107)( 88,106)( 89,105)( 90,104)( 91,125)
( 92,124)( 93,134)( 94,133)( 95,132)( 96,131)( 97,130)( 98,129)( 99,128)
(100,127)(101,126);
s3 := Sym(134)!(  3, 69)(  4, 70)(  5, 71)(  6, 72)(  7, 73)(  8, 74)(  9, 75)
( 10, 76)( 11, 77)( 12, 78)( 13, 79)( 14, 80)( 15, 81)( 16, 82)( 17, 83)
( 18, 84)( 19, 85)( 20, 86)( 21, 87)( 22, 88)( 23, 89)( 24, 90)( 25, 91)
( 26, 92)( 27, 93)( 28, 94)( 29, 95)( 30, 96)( 31, 97)( 32, 98)( 33, 99)
( 34,100)( 35,101)( 36,102)( 37,103)( 38,104)( 39,105)( 40,106)( 41,107)
( 42,108)( 43,109)( 44,110)( 45,111)( 46,112)( 47,113)( 48,114)( 49,115)
( 50,116)( 51,117)( 52,118)( 53,119)( 54,120)( 55,121)( 56,122)( 57,123)
( 58,124)( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)( 64,130)( 65,131)
( 66,132)( 67,133)( 68,134);
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*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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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