Polytope of Type {18,12,2}

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

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

Permutation Representation (Magma) :

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

to this polytope