Polytope of Type {2,2,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,6,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30782)
Rank : 5
Schlafli Type : {2,2,6,12}
Number of vertices, edges, etc : 2, 2, 18, 108, 36
Order of s₀s₁s₂s₃s₄ : 12
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 : {2,2,6,6}*864a
   3-fold quotients : {2,2,6,12}*576b
   4-fold quotients : {2,2,6,3}*432
   6-fold quotients : {2,2,6,6}*288b
   9-fold quotients : {2,2,2,12}*192
   12-fold quotients : {2,2,6,3}*144
   18-fold quotients : {2,2,2,6}*96
   27-fold quotients : {2,2,2,4}*64
   36-fold quotients : {2,2,2,3}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := (  6,  7)(  8, 11)(  9, 13)( 10, 12)( 15, 16)( 17, 20)( 18, 22)( 19, 21)
( 24, 25)( 26, 29)( 27, 31)( 28, 30)( 33, 34)( 35, 38)( 36, 40)( 37, 39)
( 42, 43)( 44, 47)( 45, 49)( 46, 48)( 51, 52)( 53, 56)( 54, 58)( 55, 57)
( 60, 61)( 62, 65)( 63, 67)( 64, 66)( 69, 70)( 71, 74)( 72, 76)( 73, 75)
( 78, 79)( 80, 83)( 81, 85)( 82, 84)( 87, 88)( 89, 92)( 90, 94)( 91, 93)
( 96, 97)( 98,101)( 99,103)(100,102)(105,106)(107,110)(108,112)(109,111);;
s3 := (  5,  8)(  6,  9)(  7, 10)( 14, 26)( 15, 27)( 16, 28)( 17, 23)( 18, 24)
( 19, 25)( 20, 29)( 21, 30)( 22, 31)( 32, 35)( 33, 36)( 34, 37)( 41, 53)
( 42, 54)( 43, 55)( 44, 50)( 45, 51)( 46, 52)( 47, 56)( 48, 57)( 49, 58)
( 59, 89)( 60, 90)( 61, 91)( 62, 86)( 63, 87)( 64, 88)( 65, 92)( 66, 93)
( 67, 94)( 68,107)( 69,108)( 70,109)( 71,104)( 72,105)( 73,106)( 74,110)
( 75,111)( 76,112)( 77, 98)( 78, 99)( 79,100)( 80, 95)( 81, 96)( 82, 97)
( 83,101)( 84,102)( 85,103);;
s4 := (  5, 68)(  6, 69)(  7, 70)(  8, 76)(  9, 74)( 10, 75)( 11, 72)( 12, 73)
( 13, 71)( 14, 59)( 15, 60)( 16, 61)( 17, 67)( 18, 65)( 19, 66)( 20, 63)
( 21, 64)( 22, 62)( 23, 77)( 24, 78)( 25, 79)( 26, 85)( 27, 83)( 28, 84)
( 29, 81)( 30, 82)( 31, 80)( 32, 95)( 33, 96)( 34, 97)( 35,103)( 36,101)
( 37,102)( 38, 99)( 39,100)( 40, 98)( 41, 86)( 42, 87)( 43, 88)( 44, 94)
( 45, 92)( 46, 93)( 47, 90)( 48, 91)( 49, 89)( 50,104)( 51,105)( 52,106)
( 53,112)( 54,110)( 55,111)( 56,108)( 57,109)( 58,107);;
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*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3*s4*s2*s3*s2*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(112)!(1,2);
s1 := Sym(112)!(3,4);
s2 := Sym(112)!(  6,  7)(  8, 11)(  9, 13)( 10, 12)( 15, 16)( 17, 20)( 18, 22)
( 19, 21)( 24, 25)( 26, 29)( 27, 31)( 28, 30)( 33, 34)( 35, 38)( 36, 40)
( 37, 39)( 42, 43)( 44, 47)( 45, 49)( 46, 48)( 51, 52)( 53, 56)( 54, 58)
( 55, 57)( 60, 61)( 62, 65)( 63, 67)( 64, 66)( 69, 70)( 71, 74)( 72, 76)
( 73, 75)( 78, 79)( 80, 83)( 81, 85)( 82, 84)( 87, 88)( 89, 92)( 90, 94)
( 91, 93)( 96, 97)( 98,101)( 99,103)(100,102)(105,106)(107,110)(108,112)
(109,111);
s3 := Sym(112)!(  5,  8)(  6,  9)(  7, 10)( 14, 26)( 15, 27)( 16, 28)( 17, 23)
( 18, 24)( 19, 25)( 20, 29)( 21, 30)( 22, 31)( 32, 35)( 33, 36)( 34, 37)
( 41, 53)( 42, 54)( 43, 55)( 44, 50)( 45, 51)( 46, 52)( 47, 56)( 48, 57)
( 49, 58)( 59, 89)( 60, 90)( 61, 91)( 62, 86)( 63, 87)( 64, 88)( 65, 92)
( 66, 93)( 67, 94)( 68,107)( 69,108)( 70,109)( 71,104)( 72,105)( 73,106)
( 74,110)( 75,111)( 76,112)( 77, 98)( 78, 99)( 79,100)( 80, 95)( 81, 96)
( 82, 97)( 83,101)( 84,102)( 85,103);
s4 := Sym(112)!(  5, 68)(  6, 69)(  7, 70)(  8, 76)(  9, 74)( 10, 75)( 11, 72)
( 12, 73)( 13, 71)( 14, 59)( 15, 60)( 16, 61)( 17, 67)( 18, 65)( 19, 66)
( 20, 63)( 21, 64)( 22, 62)( 23, 77)( 24, 78)( 25, 79)( 26, 85)( 27, 83)
( 28, 84)( 29, 81)( 30, 82)( 31, 80)( 32, 95)( 33, 96)( 34, 97)( 35,103)
( 36,101)( 37,102)( 38, 99)( 39,100)( 40, 98)( 41, 86)( 42, 87)( 43, 88)
( 44, 94)( 45, 92)( 46, 93)( 47, 90)( 48, 91)( 49, 89)( 50,104)( 51,105)
( 52,106)( 53,112)( 54,110)( 55,111)( 56,108)( 57,109)( 58,107);
poly := sub<Sym(112)|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*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3*s4*s2*s3*s2*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

to this polytope