Polytope of Type {6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3}*1296
Also Known As : {6,3}_(6,6). if this polytope has another name.
Group : SmallGroup(1296,1784)
Rank : 3
Schlafli Type : {6,3}
Number of vertices, edges, etc : 216, 324, 108
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,3}*432
   4-fold quotients : {6,3}*324
   9-fold quotients : {6,3}*144
   12-fold quotients : {6,3}*108
   27-fold quotients : {6,3}*48
   36-fold quotients : {6,3}*36
   54-fold quotients : {3,3}*24
   108-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

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

References : None.
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