Polytope of Type {2,12,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,18}*864c
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 4
Schlafli Type : {2,12,18}
Number of vertices, edges, etc : 2, 12, 108, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,12,18,2} of size 1728
Vertex Figure Of :
   {2,2,12,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,4,18}*288c, {2,12,6}*288d
   6-fold quotients : {2,4,9}*144
   9-fold quotients : {2,4,6}*96b
   18-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,12,18}*1728a
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3 >;

to this polytope