Polytope of Type {26,4,4,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {26,4,4,2}*1664
if this polytope has a name.
Group : SmallGroup(1664,17724)
Rank : 5
Schlafli Type : {26,4,4,2}
Number of vertices, edges, etc : 26, 52, 8, 4, 2
Order of s₀s₁s₂s₃s₄ : 52
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 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 : {26,2,4,2}*832, {26,4,2,2}*832
   4-fold quotients : {13,2,4,2}*416, {26,2,2,2}*416
   8-fold quotients : {13,2,2,2}*208
   13-fold quotients : {2,4,4,2}*128
   26-fold quotients : {2,2,4,2}*64, {2,4,2,2}*64
   52-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 15, 26)( 16, 25)
( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 28, 39)( 29, 38)( 30, 37)( 31, 36)
( 32, 35)( 33, 34)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)( 46, 47)
( 54, 65)( 55, 64)( 56, 63)( 57, 62)( 58, 61)( 59, 60)( 67, 78)( 68, 77)
( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 80, 91)( 81, 90)( 82, 89)( 83, 88)
( 84, 87)( 85, 86)( 93,104)( 94,103)( 95,102)( 96,101)( 97,100)( 98, 99);;
s1 := (  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 15)( 16, 26)
( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 28)( 29, 39)( 30, 38)( 31, 37)
( 32, 36)( 33, 35)( 40, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)( 46, 48)
( 53, 67)( 54, 66)( 55, 78)( 56, 77)( 57, 76)( 58, 75)( 59, 74)( 60, 73)
( 61, 72)( 62, 71)( 63, 70)( 64, 69)( 65, 68)( 79, 93)( 80, 92)( 81,104)
( 82,103)( 83,102)( 84,101)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89, 96)
( 90, 95)( 91, 94);;
s2 := (  1, 53)(  2, 54)(  3, 55)(  4, 56)(  5, 57)(  6, 58)(  7, 59)(  8, 60)
(  9, 61)( 10, 62)( 11, 63)( 12, 64)( 13, 65)( 14, 66)( 15, 67)( 16, 68)
( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 73)( 22, 74)( 23, 75)( 24, 76)
( 25, 77)( 26, 78)( 27, 79)( 28, 80)( 29, 81)( 30, 82)( 31, 83)( 32, 84)
( 33, 85)( 34, 86)( 35, 87)( 36, 88)( 37, 89)( 38, 90)( 39, 91)( 40, 92)
( 41, 93)( 42, 94)( 43, 95)( 44, 96)( 45, 97)( 46, 98)( 47, 99)( 48,100)
( 49,101)( 50,102)( 51,103)( 52,104);;
s3 := ( 53, 79)( 54, 80)( 55, 81)( 56, 82)( 57, 83)( 58, 84)( 59, 85)( 60, 86)
( 61, 87)( 62, 88)( 63, 89)( 64, 90)( 65, 91)( 66, 92)( 67, 93)( 68, 94)
( 69, 95)( 70, 96)( 71, 97)( 72, 98)( 73, 99)( 74,100)( 75,101)( 76,102)
( 77,103)( 78,104);;
s4 := (105,106);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(106)!(  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 15, 26)
( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 28, 39)( 29, 38)( 30, 37)
( 31, 36)( 32, 35)( 33, 34)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 54, 65)( 55, 64)( 56, 63)( 57, 62)( 58, 61)( 59, 60)( 67, 78)
( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 80, 91)( 81, 90)( 82, 89)
( 83, 88)( 84, 87)( 85, 86)( 93,104)( 94,103)( 95,102)( 96,101)( 97,100)
( 98, 99);
s1 := Sym(106)!(  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 15)
( 16, 26)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 28)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 40, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)
( 46, 48)( 53, 67)( 54, 66)( 55, 78)( 56, 77)( 57, 76)( 58, 75)( 59, 74)
( 60, 73)( 61, 72)( 62, 71)( 63, 70)( 64, 69)( 65, 68)( 79, 93)( 80, 92)
( 81,104)( 82,103)( 83,102)( 84,101)( 85,100)( 86, 99)( 87, 98)( 88, 97)
( 89, 96)( 90, 95)( 91, 94);
s2 := Sym(106)!(  1, 53)(  2, 54)(  3, 55)(  4, 56)(  5, 57)(  6, 58)(  7, 59)
(  8, 60)(  9, 61)( 10, 62)( 11, 63)( 12, 64)( 13, 65)( 14, 66)( 15, 67)
( 16, 68)( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 73)( 22, 74)( 23, 75)
( 24, 76)( 25, 77)( 26, 78)( 27, 79)( 28, 80)( 29, 81)( 30, 82)( 31, 83)
( 32, 84)( 33, 85)( 34, 86)( 35, 87)( 36, 88)( 37, 89)( 38, 90)( 39, 91)
( 40, 92)( 41, 93)( 42, 94)( 43, 95)( 44, 96)( 45, 97)( 46, 98)( 47, 99)
( 48,100)( 49,101)( 50,102)( 51,103)( 52,104);
s3 := Sym(106)!( 53, 79)( 54, 80)( 55, 81)( 56, 82)( 57, 83)( 58, 84)( 59, 85)
( 60, 86)( 61, 87)( 62, 88)( 63, 89)( 64, 90)( 65, 91)( 66, 92)( 67, 93)
( 68, 94)( 69, 95)( 70, 96)( 71, 97)( 72, 98)( 73, 99)( 74,100)( 75,101)
( 76,102)( 77,103)( 78,104);
s4 := Sym(106)!(105,106);
poly := sub<Sym(106)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

to this polytope