Polytope of Type {6,9,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9,4}*432
if this polytope has a name.
Group : SmallGroup(432,522)
Rank : 4
Schlafli Type : {6,9,4}
Number of vertices, edges, etc : 6, 27, 18, 4
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,9,4,2} of size 864
Vertex Figure Of :
   {2,6,9,4} of size 864
   {3,6,9,4} of size 1296
   {4,6,9,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,9,4}*144, {6,3,4}*144
   9-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,9,4}*864, {6,18,4}*864d, {6,18,4}*864e
   3-fold covers : {6,27,4}*1296, {18,9,4}*1296, {6,9,4}*1296c
   4-fold covers : {6,36,4}*1728e, {6,36,4}*1728f, {6,9,8}*1728, {12,18,4}*1728d, {6,9,4}*1728, {6,18,4}*1728b, {12,9,4}*1728
Permutation Representation (GAP) :

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

References : None.
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