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}*1728c
if this polytope has a name.
Group : SmallGroup(1728,30882)
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}*864c
   3-fold quotients : {2,2,6,12}*576c
   4-fold quotients : {2,2,3,6}*432
   6-fold quotients : {2,2,6,6}*288c
   9-fold quotients : {2,2,6,4}*192a
   12-fold quotients : {2,2,3,6}*144
   18-fold quotients : {2,2,6,2}*96
   27-fold quotients : {2,2,2,4}*64
   36-fold quotients : {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 := (  8, 11)(  9, 12)( 10, 13)( 14, 23)( 15, 24)( 16, 25)( 17, 29)( 18, 30)
( 19, 31)( 20, 26)( 21, 27)( 22, 28)( 35, 38)( 36, 39)( 37, 40)( 41, 50)
( 42, 51)( 43, 52)( 44, 56)( 45, 57)( 46, 58)( 47, 53)( 48, 54)( 49, 55)
( 62, 65)( 63, 66)( 64, 67)( 68, 77)( 69, 78)( 70, 79)( 71, 83)( 72, 84)
( 73, 85)( 74, 80)( 75, 81)( 76, 82)( 89, 92)( 90, 93)( 91, 94)( 95,104)
( 96,105)( 97,106)( 98,110)( 99,111)(100,112)(101,107)(102,108)(103,109);;
s3 := (  5, 81)(  6, 82)(  7, 80)(  8, 79)(  9, 77)( 10, 78)( 11, 83)( 12, 84)
( 13, 85)( 14, 72)( 15, 73)( 16, 71)( 17, 70)( 18, 68)( 19, 69)( 20, 74)
( 21, 75)( 22, 76)( 23, 63)( 24, 64)( 25, 62)( 26, 61)( 27, 59)( 28, 60)
( 29, 65)( 30, 66)( 31, 67)( 32,108)( 33,109)( 34,107)( 35,106)( 36,104)
( 37,105)( 38,110)( 39,111)( 40,112)( 41, 99)( 42,100)( 43, 98)( 44, 97)
( 45, 95)( 46, 96)( 47,101)( 48,102)( 49,103)( 50, 90)( 51, 91)( 52, 89)
( 53, 88)( 54, 86)( 55, 87)( 56, 92)( 57, 93)( 58, 94);;
s4 := (  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)
( 59, 86)( 60, 88)( 61, 87)( 62, 92)( 63, 94)( 64, 93)( 65, 89)( 66, 91)
( 67, 90)( 68, 95)( 69, 97)( 70, 96)( 71,101)( 72,103)( 73,102)( 74, 98)
( 75,100)( 76, 99)( 77,104)( 78,106)( 79,105)( 80,110)( 81,112)( 82,111)
( 83,107)( 84,109)( 85,108);;
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, 
s4*s2*s3*s4*s3*s2*s3*s2*s3*s2*s4*s3*s4*s3*s2*s3*s2*s3, 
s4*s2*s3*s4*s3*s4*s3*s4*s3*s4*s2*s3*s2*s3*s4*s3*s2*s3, 
s4*s2*s3*s4*s3*s2*s3*s4*s3*s4*s2*s3*s4*s3*s2*s3*s4*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope