Polytope of Type {2,18,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,4,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,30790)
Rank : 5
Schlafli Type : {2,18,4,6}
Number of vertices, edges, etc : 2, 18, 36, 12, 6
Order of s₀s₁s₂s₃s₄ : 36
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,18,2,6}*864
   3-fold quotients : {2,18,4,2}*576a, {2,6,4,6}*576
   4-fold quotients : {2,9,2,6}*432, {2,18,2,3}*432
   6-fold quotients : {2,18,2,2}*288, {2,6,2,6}*288
   8-fold quotients : {2,9,2,3}*216
   9-fold quotients : {2,2,4,6}*192a, {2,6,4,2}*192a
   12-fold quotients : {2,9,2,2}*144, {2,3,2,6}*144, {2,6,2,3}*144
   18-fold quotients : {2,2,2,6}*96, {2,6,2,2}*96
   24-fold quotients : {2,3,2,3}*72
   27-fold quotients : {2,2,4,2}*64
   36-fold quotients : {2,2,2,3}*48, {2,3,2,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  7,  8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)( 16, 25)
( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 34, 35)( 37, 38)( 39, 50)
( 40, 49)( 41, 48)( 42, 53)( 43, 52)( 44, 51)( 45, 56)( 46, 55)( 47, 54)
( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)( 68, 75)( 69, 80)( 70, 79)
( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 85, 86)( 88, 89)( 91, 92)( 93,104)
( 94,103)( 95,102)( 96,107)( 97,106)( 98,105)( 99,110)(100,109)(101,108);;
s2 := (  3, 12)(  4, 14)(  5, 13)(  6, 15)(  7, 17)(  8, 16)(  9, 18)( 10, 20)
( 11, 19)( 21, 23)( 24, 26)( 27, 29)( 30, 39)( 31, 41)( 32, 40)( 33, 42)
( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 48, 50)( 51, 53)( 54, 56)
( 57, 93)( 58, 95)( 59, 94)( 60, 96)( 61, 98)( 62, 97)( 63, 99)( 64,101)
( 65,100)( 66, 84)( 67, 86)( 68, 85)( 69, 87)( 70, 89)( 71, 88)( 72, 90)
( 73, 92)( 74, 91)( 75,104)( 76,103)( 77,102)( 78,107)( 79,106)( 80,105)
( 81,110)( 82,109)( 83,108);;
s3 := (  3, 57)(  4, 58)(  5, 59)(  6, 63)(  7, 64)(  8, 65)(  9, 60)( 10, 61)
( 11, 62)( 12, 66)( 13, 67)( 14, 68)( 15, 72)( 16, 73)( 17, 74)( 18, 69)
( 19, 70)( 20, 71)( 21, 75)( 22, 76)( 23, 77)( 24, 81)( 25, 82)( 26, 83)
( 27, 78)( 28, 79)( 29, 80)( 30, 84)( 31, 85)( 32, 86)( 33, 90)( 34, 91)
( 35, 92)( 36, 87)( 37, 88)( 38, 89)( 39, 93)( 40, 94)( 41, 95)( 42, 99)
( 43,100)( 44,101)( 45, 96)( 46, 97)( 47, 98)( 48,102)( 49,103)( 50,104)
( 51,108)( 52,109)( 53,110)( 54,105)( 55,106)( 56,107);;
s4 := (  3,  6)(  4,  7)(  5,  8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)( 22, 25)
( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)( 48, 51)
( 49, 52)( 50, 53)( 57, 60)( 58, 61)( 59, 62)( 66, 69)( 67, 70)( 68, 71)
( 75, 78)( 76, 79)( 77, 80)( 84, 87)( 85, 88)( 86, 89)( 93, 96)( 94, 97)
( 95, 98)(102,105)(103,106)(104,107);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
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(110)!(1,2);
s1 := Sym(110)!(  4,  5)(  7,  8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)
( 16, 25)( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 34, 35)( 37, 38)
( 39, 50)( 40, 49)( 41, 48)( 42, 53)( 43, 52)( 44, 51)( 45, 56)( 46, 55)
( 47, 54)( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)( 68, 75)( 69, 80)
( 70, 79)( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 85, 86)( 88, 89)( 91, 92)
( 93,104)( 94,103)( 95,102)( 96,107)( 97,106)( 98,105)( 99,110)(100,109)
(101,108);
s2 := Sym(110)!(  3, 12)(  4, 14)(  5, 13)(  6, 15)(  7, 17)(  8, 16)(  9, 18)
( 10, 20)( 11, 19)( 21, 23)( 24, 26)( 27, 29)( 30, 39)( 31, 41)( 32, 40)
( 33, 42)( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 48, 50)( 51, 53)
( 54, 56)( 57, 93)( 58, 95)( 59, 94)( 60, 96)( 61, 98)( 62, 97)( 63, 99)
( 64,101)( 65,100)( 66, 84)( 67, 86)( 68, 85)( 69, 87)( 70, 89)( 71, 88)
( 72, 90)( 73, 92)( 74, 91)( 75,104)( 76,103)( 77,102)( 78,107)( 79,106)
( 80,105)( 81,110)( 82,109)( 83,108);
s3 := Sym(110)!(  3, 57)(  4, 58)(  5, 59)(  6, 63)(  7, 64)(  8, 65)(  9, 60)
( 10, 61)( 11, 62)( 12, 66)( 13, 67)( 14, 68)( 15, 72)( 16, 73)( 17, 74)
( 18, 69)( 19, 70)( 20, 71)( 21, 75)( 22, 76)( 23, 77)( 24, 81)( 25, 82)
( 26, 83)( 27, 78)( 28, 79)( 29, 80)( 30, 84)( 31, 85)( 32, 86)( 33, 90)
( 34, 91)( 35, 92)( 36, 87)( 37, 88)( 38, 89)( 39, 93)( 40, 94)( 41, 95)
( 42, 99)( 43,100)( 44,101)( 45, 96)( 46, 97)( 47, 98)( 48,102)( 49,103)
( 50,104)( 51,108)( 52,109)( 53,110)( 54,105)( 55,106)( 56,107);
s4 := Sym(110)!(  3,  6)(  4,  7)(  5,  8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)
( 22, 25)( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)
( 48, 51)( 49, 52)( 50, 53)( 57, 60)( 58, 61)( 59, 62)( 66, 69)( 67, 70)
( 68, 71)( 75, 78)( 76, 79)( 77, 80)( 84, 87)( 85, 88)( 86, 89)( 93, 96)
( 94, 97)( 95, 98)(102,105)(103,106)(104,107);
poly := sub<Sym(110)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
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 >;

to this polytope