Polytope of Type {2,18,12}

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

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

Permutation Representation (Magma) :

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

to this polytope