Polytope of Type {4,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,6}*864a
Also Known As : {{4,18|2},{18,6|2}}. if this polytope has another name.
Group : SmallGroup(864,2462)
Rank : 4
Schlafli Type : {4,18,6}
Number of vertices, edges, etc : 4, 36, 54, 6
Order of s₀s₁s₂s₃ : 36
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,18,6,2} of size 1728
Vertex Figure Of :
   {2,4,18,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,18,6}*432a
   3-fold quotients : {4,18,2}*288a, {4,6,6}*288a
   6-fold quotients : {2,18,2}*144, {2,6,6}*144a
   9-fold quotients : {4,2,6}*96, {4,6,2}*96a
   12-fold quotients : {2,9,2}*72
   18-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
   27-fold quotients : {4,2,2}*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 : {4,18,12}*1728a, {4,36,6}*1728a, {8,18,6}*1728a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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