Polytope of Type {2,12,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,18}*864a
if this polytope has a name.
Group : SmallGroup(864,2438)
Rank : 4
Schlafli Type : {2,12,18}
Number of vertices, edges, etc : 2, 12, 108, 18
Order of s₀s₁s₂s₃ : 36
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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) :
   2-fold quotients : {2,6,18}*432a
   3-fold quotients : {2,4,18}*288a, {2,12,6}*288a
   6-fold quotients : {2,2,18}*144, {2,6,6}*144a
   9-fold quotients : {2,12,2}*96, {2,4,6}*96a
   12-fold quotients : {2,2,9}*72
   18-fold quotients : {2,2,6}*48, {2,6,2}*48
   27-fold quotients : {2,4,2}*32
   36-fold quotients : {2,2,3}*24, {2,3,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12,18}*1728a, {2,24,18}*1728a, {2,12,36}*1728a
Permutation Representation (GAP) :

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

to this polytope