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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,6,2}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30782)
Rank : 5
Schlafli Type : {2,12,6,2}
Number of vertices, edges, etc : 2, 36, 108, 18, 2
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 : {2,6,6,2}*864b
   3-fold quotients : {2,12,6,2}*576a
   4-fold quotients : {2,6,6,2}*432
   6-fold quotients : {2,6,6,2}*288a
   9-fold quotients : {2,12,2,2}*192, {2,4,6,2}*192a
   18-fold quotients : {2,2,6,2}*96, {2,6,2,2}*96
   27-fold quotients : {2,4,2,2}*64
   36-fold quotients : {2,2,3,2}*48, {2,3,2,2}*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 := (  4,  5)(  7,  8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)
( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)( 47, 55)
( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)( 64, 92)
( 65, 91)( 66,102)( 67,104)( 68,103)( 69,105)( 70,107)( 71,106)( 72,108)
( 73,110)( 74,109)( 75, 93)( 76, 95)( 77, 94)( 78, 96)( 79, 98)( 80, 97)
( 81, 99)( 82,101)( 83,100);;
s2 := (  3, 66)(  4, 67)(  5, 68)(  6, 74)(  7, 72)(  8, 73)(  9, 70)( 10, 71)
( 11, 69)( 12, 57)( 13, 58)( 14, 59)( 15, 65)( 16, 63)( 17, 64)( 18, 61)
( 19, 62)( 20, 60)( 21, 75)( 22, 76)( 23, 77)( 24, 83)( 25, 81)( 26, 82)
( 27, 79)( 28, 80)( 29, 78)( 30, 93)( 31, 94)( 32, 95)( 33,101)( 34, 99)
( 35,100)( 36, 97)( 37, 98)( 38, 96)( 39, 84)( 40, 85)( 41, 86)( 42, 92)
( 43, 90)( 44, 91)( 45, 88)( 46, 89)( 47, 87)( 48,102)( 49,103)( 50,104)
( 51,110)( 52,108)( 53,109)( 54,106)( 55,107)( 56,105);;
s3 := (  3,  6)(  4,  8)(  5,  7)( 10, 11)( 12, 15)( 13, 17)( 14, 16)( 19, 20)
( 21, 24)( 22, 26)( 23, 25)( 28, 29)( 30, 33)( 31, 35)( 32, 34)( 37, 38)
( 39, 42)( 40, 44)( 41, 43)( 46, 47)( 48, 51)( 49, 53)( 50, 52)( 55, 56)
( 57, 60)( 58, 62)( 59, 61)( 64, 65)( 66, 69)( 67, 71)( 68, 70)( 73, 74)
( 75, 78)( 76, 80)( 77, 79)( 82, 83)( 84, 87)( 85, 89)( 86, 88)( 91, 92)
( 93, 96)( 94, 98)( 95, 97)(100,101)(102,105)(103,107)(104,106)(109,110);;
s4 := (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, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2, 
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)(  7,  8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)
( 16, 26)( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)
( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)
( 47, 55)( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)
( 64, 92)( 65, 91)( 66,102)( 67,104)( 68,103)( 69,105)( 70,107)( 71,106)
( 72,108)( 73,110)( 74,109)( 75, 93)( 76, 95)( 77, 94)( 78, 96)( 79, 98)
( 80, 97)( 81, 99)( 82,101)( 83,100);
s2 := Sym(112)!(  3, 66)(  4, 67)(  5, 68)(  6, 74)(  7, 72)(  8, 73)(  9, 70)
( 10, 71)( 11, 69)( 12, 57)( 13, 58)( 14, 59)( 15, 65)( 16, 63)( 17, 64)
( 18, 61)( 19, 62)( 20, 60)( 21, 75)( 22, 76)( 23, 77)( 24, 83)( 25, 81)
( 26, 82)( 27, 79)( 28, 80)( 29, 78)( 30, 93)( 31, 94)( 32, 95)( 33,101)
( 34, 99)( 35,100)( 36, 97)( 37, 98)( 38, 96)( 39, 84)( 40, 85)( 41, 86)
( 42, 92)( 43, 90)( 44, 91)( 45, 88)( 46, 89)( 47, 87)( 48,102)( 49,103)
( 50,104)( 51,110)( 52,108)( 53,109)( 54,106)( 55,107)( 56,105);
s3 := Sym(112)!(  3,  6)(  4,  8)(  5,  7)( 10, 11)( 12, 15)( 13, 17)( 14, 16)
( 19, 20)( 21, 24)( 22, 26)( 23, 25)( 28, 29)( 30, 33)( 31, 35)( 32, 34)
( 37, 38)( 39, 42)( 40, 44)( 41, 43)( 46, 47)( 48, 51)( 49, 53)( 50, 52)
( 55, 56)( 57, 60)( 58, 62)( 59, 61)( 64, 65)( 66, 69)( 67, 71)( 68, 70)
( 73, 74)( 75, 78)( 76, 80)( 77, 79)( 82, 83)( 84, 87)( 85, 89)( 86, 88)
( 91, 92)( 93, 96)( 94, 98)( 95, 97)(100,101)(102,105)(103,107)(104,106)
(109,110);
s4 := Sym(112)!(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, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2, 
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