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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,12,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30782)
Rank : 5
Schlafli Type : {2,2,12,6}
Number of vertices, edges, etc : 2, 2, 36, 108, 18
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}*864b
   3-fold quotients : {2,2,12,6}*576a
   4-fold quotients : {2,2,6,6}*432
   6-fold quotients : {2,2,6,6}*288a
   9-fold quotients : {2,2,12,2}*192, {2,2,4,6}*192a
   18-fold quotients : {2,2,2,6}*96, {2,2,6,2}*96
   27-fold quotients : {2,2,4,2}*64
   36-fold quotients : {2,2,2,3}*48, {2,2,3,2}*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)(  9, 10)( 12, 13)( 14, 23)( 15, 25)( 16, 24)( 17, 26)( 18, 28)
( 19, 27)( 20, 29)( 21, 31)( 22, 30)( 33, 34)( 36, 37)( 39, 40)( 41, 50)
( 42, 52)( 43, 51)( 44, 53)( 45, 55)( 46, 54)( 47, 56)( 48, 58)( 49, 57)
( 59, 86)( 60, 88)( 61, 87)( 62, 89)( 63, 91)( 64, 90)( 65, 92)( 66, 94)
( 67, 93)( 68,104)( 69,106)( 70,105)( 71,107)( 72,109)( 73,108)( 74,110)
( 75,112)( 76,111)( 77, 95)( 78, 97)( 79, 96)( 80, 98)( 81,100)( 82, 99)
( 83,101)( 84,103)( 85,102);;
s3 := (  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);;
s4 := (  5,  8)(  6, 10)(  7,  9)( 12, 13)( 14, 17)( 15, 19)( 16, 18)( 21, 22)
( 23, 26)( 24, 28)( 25, 27)( 30, 31)( 32, 35)( 33, 37)( 34, 36)( 39, 40)
( 41, 44)( 42, 46)( 43, 45)( 48, 49)( 50, 53)( 51, 55)( 52, 54)( 57, 58)
( 59, 62)( 60, 64)( 61, 63)( 66, 67)( 68, 71)( 69, 73)( 70, 72)( 75, 76)
( 77, 80)( 78, 82)( 79, 81)( 84, 85)( 86, 89)( 87, 91)( 88, 90)( 93, 94)
( 95, 98)( 96,100)( 97, 99)(102,103)(104,107)(105,109)(106,108)(111,112);;
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, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s4*s3*s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope