Polytope of Type {2,6,9,4,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,9,4,2}*1728
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 6
Schlafli Type : {2,6,9,4,2}
Number of vertices, edges, etc : 2, 6, 27, 18, 4, 2
Order of s0s1s2s3s4s5 : 18
Order of s0s1s2s3s4s5s4s3s2s1 : 2
Special Properties :
   Degenerate
   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 : {2,2,9,4,2}*576, {2,6,3,4,2}*576
   9-fold quotients : {2,2,3,4,2}*192
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)( 22, 34)
( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)( 54, 66)
( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)( 62, 74)
( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)
( 95,107)( 96,108)( 97,109)( 98,110);;
s2 := (  3, 15)(  4, 17)(  5, 16)(  6, 18)(  7, 23)(  8, 25)(  9, 24)( 10, 26)
( 11, 19)( 12, 21)( 13, 20)( 14, 22)( 28, 29)( 31, 35)( 32, 37)( 33, 36)
( 34, 38)( 39, 91)( 40, 93)( 41, 92)( 42, 94)( 43, 87)( 44, 89)( 45, 88)
( 46, 90)( 47, 95)( 48, 97)( 49, 96)( 50, 98)( 51, 79)( 52, 81)( 53, 80)
( 54, 82)( 55, 75)( 56, 77)( 57, 76)( 58, 78)( 59, 83)( 60, 85)( 61, 84)
( 62, 86)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)
( 70,102)( 71,107)( 72,109)( 73,108)( 74,110);;
s3 := (  3, 39)(  4, 40)(  5, 42)(  6, 41)(  7, 47)(  8, 48)(  9, 50)( 10, 49)
( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 63)( 16, 64)( 17, 66)( 18, 65)
( 19, 71)( 20, 72)( 21, 74)( 22, 73)( 23, 67)( 24, 68)( 25, 70)( 26, 69)
( 27, 51)( 28, 52)( 29, 54)( 30, 53)( 31, 59)( 32, 60)( 33, 62)( 34, 61)
( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 75, 79)( 76, 80)( 77, 82)( 78, 81)
( 85, 86)( 87,103)( 88,104)( 89,106)( 90,105)( 91, 99)( 92,100)( 93,102)
( 94,101)( 95,107)( 96,108)( 97,110)( 98,109);;
s4 := (  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)( 16, 17)
( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)( 32, 33)
( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)
( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)( 64, 65)
( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)( 80, 81)
( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)( 96, 97)
( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109);;
s5 := (111,112);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
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, s0*s5*s0*s5, s1*s5*s1*s5, 
s2*s5*s2*s5, s3*s5*s3*s5, s4*s5*s4*s5, 
s3*s4*s3*s4*s3*s4*s3*s4, s4*s3*s2*s4*s3*s4*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(112)!(1,2);
s1 := Sym(112)!( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)
( 22, 34)( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)
( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)
( 62, 74)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)
( 94,106)( 95,107)( 96,108)( 97,109)( 98,110);
s2 := Sym(112)!(  3, 15)(  4, 17)(  5, 16)(  6, 18)(  7, 23)(  8, 25)(  9, 24)
( 10, 26)( 11, 19)( 12, 21)( 13, 20)( 14, 22)( 28, 29)( 31, 35)( 32, 37)
( 33, 36)( 34, 38)( 39, 91)( 40, 93)( 41, 92)( 42, 94)( 43, 87)( 44, 89)
( 45, 88)( 46, 90)( 47, 95)( 48, 97)( 49, 96)( 50, 98)( 51, 79)( 52, 81)
( 53, 80)( 54, 82)( 55, 75)( 56, 77)( 57, 76)( 58, 78)( 59, 83)( 60, 85)
( 61, 84)( 62, 86)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)
( 69,100)( 70,102)( 71,107)( 72,109)( 73,108)( 74,110);
s3 := Sym(112)!(  3, 39)(  4, 40)(  5, 42)(  6, 41)(  7, 47)(  8, 48)(  9, 50)
( 10, 49)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 63)( 16, 64)( 17, 66)
( 18, 65)( 19, 71)( 20, 72)( 21, 74)( 22, 73)( 23, 67)( 24, 68)( 25, 70)
( 26, 69)( 27, 51)( 28, 52)( 29, 54)( 30, 53)( 31, 59)( 32, 60)( 33, 62)
( 34, 61)( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 75, 79)( 76, 80)( 77, 82)
( 78, 81)( 85, 86)( 87,103)( 88,104)( 89,106)( 90,105)( 91, 99)( 92,100)
( 93,102)( 94,101)( 95,107)( 96,108)( 97,110)( 98,109);
s4 := Sym(112)!(  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)
( 16, 17)( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)
( 32, 33)( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)
( 48, 49)( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)
( 64, 65)( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)
( 80, 81)( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)
( 96, 97)( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109);
s5 := Sym(112)!(111,112);
poly := sub<Sym(112)|s0,s1,s2,s3,s4,s5>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, 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, s0*s5*s0*s5, 
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5, 
s4*s5*s4*s5, s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s3*s2*s4*s3*s4*s3*s2*s3, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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