Polytope of Type {6,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,27}*1296
if this polytope has a name.
Group : SmallGroup(1296,1780)
Rank : 3
Schlafli Type : {6,27}
Number of vertices, edges, etc : 24, 324, 108
Order of s₀s₁s₂ : 108
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,9}*432
   4-fold quotients : {6,27}*324
   9-fold quotients : {6,3}*144
   12-fold quotients : {2,27}*108, {6,9}*108
   27-fold quotients : {6,3}*48
   36-fold quotients : {2,9}*36, {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)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 63, 64)
( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 83, 84)( 87, 88)( 91, 92)( 95, 96)
( 99,100)(103,104)(107,108);;
s1 := (  2,  4)(  5,  9)(  6, 12)(  7, 11)(  8, 10)( 13, 33)( 14, 36)( 15, 35)
( 16, 34)( 17, 29)( 18, 32)( 19, 31)( 20, 30)( 21, 25)( 22, 28)( 23, 27)
( 24, 26)( 37,105)( 38,108)( 39,107)( 40,106)( 41,101)( 42,104)( 43,103)
( 44,102)( 45, 97)( 46,100)( 47, 99)( 48, 98)( 49, 93)( 50, 96)( 51, 95)
( 52, 94)( 53, 89)( 54, 92)( 55, 91)( 56, 90)( 57, 85)( 58, 88)( 59, 87)
( 60, 86)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 77)( 66, 80)( 67, 79)
( 68, 78)( 69, 73)( 70, 76)( 71, 75)( 72, 74);;
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, 70)( 14, 69)( 15, 71)( 16, 72)
( 17, 66)( 18, 65)( 19, 67)( 20, 68)( 21, 62)( 22, 61)( 23, 63)( 24, 64)
( 25, 58)( 26, 57)( 27, 59)( 28, 60)( 29, 54)( 30, 53)( 31, 55)( 32, 56)
( 33, 50)( 34, 49)( 35, 51)( 36, 52)( 73,106)( 74,105)( 75,107)( 76,108)
( 77,102)( 78,101)( 79,103)( 80,104)( 81, 98)( 82, 97)( 83, 99)( 84,100)
( 85, 94)( 86, 93)( 87, 95)( 88, 96)( 89, 90);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
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*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)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)
( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 83, 84)( 87, 88)( 91, 92)
( 95, 96)( 99,100)(103,104)(107,108);
s1 := Sym(108)!(  2,  4)(  5,  9)(  6, 12)(  7, 11)(  8, 10)( 13, 33)( 14, 36)
( 15, 35)( 16, 34)( 17, 29)( 18, 32)( 19, 31)( 20, 30)( 21, 25)( 22, 28)
( 23, 27)( 24, 26)( 37,105)( 38,108)( 39,107)( 40,106)( 41,101)( 42,104)
( 43,103)( 44,102)( 45, 97)( 46,100)( 47, 99)( 48, 98)( 49, 93)( 50, 96)
( 51, 95)( 52, 94)( 53, 89)( 54, 92)( 55, 91)( 56, 90)( 57, 85)( 58, 88)
( 59, 87)( 60, 86)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 77)( 66, 80)
( 67, 79)( 68, 78)( 69, 73)( 70, 76)( 71, 75)( 72, 74);
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, 70)( 14, 69)( 15, 71)
( 16, 72)( 17, 66)( 18, 65)( 19, 67)( 20, 68)( 21, 62)( 22, 61)( 23, 63)
( 24, 64)( 25, 58)( 26, 57)( 27, 59)( 28, 60)( 29, 54)( 30, 53)( 31, 55)
( 32, 56)( 33, 50)( 34, 49)( 35, 51)( 36, 52)( 73,106)( 74,105)( 75,107)
( 76,108)( 77,102)( 78,101)( 79,103)( 80,104)( 81, 98)( 82, 97)( 83, 99)
( 84,100)( 85, 94)( 86, 93)( 87, 95)( 88, 96)( 89, 90);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
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