Polytope of Type {2,2,9,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,9,12}*1728
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 5
Schlafli Type : {2,2,9,12}
Number of vertices, edges, etc : 2, 2, 18, 108, 24
Order of s₀s₁s₂s₃s₄ : 18
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) :
   3-fold quotients : {2,2,9,4}*576, {2,2,3,12}*576
   4-fold quotients : {2,2,9,6}*432
   6-fold quotients : {2,2,9,4}*288
   9-fold quotients : {2,2,3,4}*192
   12-fold quotients : {2,2,9,2}*144, {2,2,3,6}*144
   18-fold quotients : {2,2,3,4}*96
   36-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

to this polytope