Polytope of Type {2,108,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,108,2}*864
if this polytope has a name.
Group : SmallGroup(864,633)
Rank : 4
Schlafli Type : {2,108,2}
Number of vertices, edges, etc : 2, 108, 108, 2
Order of s₀s₁s₂s₃ : 108
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,108,2,2} of size 1728
Vertex Figure Of :
   {2,2,108,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,54,2}*432
   3-fold quotients : {2,36,2}*288
   4-fold quotients : {2,27,2}*216
   6-fold quotients : {2,18,2}*144
   9-fold quotients : {2,12,2}*96
   12-fold quotients : {2,9,2}*72
   18-fold quotients : {2,6,2}*48
   27-fold quotients : {2,4,2}*32
   36-fold quotients : {2,3,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,108,4}*1728a, {4,108,2}*1728a, {2,216,2}*1728
Permutation Representation (GAP) :

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