Polytope of Type {18,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,4}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1785)
Rank : 4
Schlafli Type : {18,6,4}
Number of vertices, edges, etc : 27, 81, 18, 4
Order of s₀s₁s₂s₃ : 9
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-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) :
   3-fold quotients : {6,6,4}*432
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope