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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,9,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 5
Schlafli Type : {6,2,9,4}
Number of vertices, edges, etc : 6, 6, 18, 36, 8
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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) :
   2-fold quotients : {3,2,9,4}*864, {6,2,9,4}*864
   3-fold quotients : {2,2,9,4}*576, {6,2,3,4}*576
   4-fold quotients : {3,2,9,4}*432, {6,2,9,2}*432
   6-fold quotients : {2,2,9,4}*288, {3,2,3,4}*288, {6,2,3,4}*288
   8-fold quotients : {3,2,9,2}*216
   9-fold quotients : {2,2,3,4}*192
   12-fold quotients : {2,2,9,2}*144, {3,2,3,4}*144, {6,2,3,2}*144
   18-fold quotients : {2,2,3,4}*96
   24-fold quotients : {3,2,3,2}*72
   36-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := ( 8, 9)(11,15)(12,17)(13,16)(14,18)(19,35)(20,37)(21,36)(22,38)(23,31)
(24,33)(25,32)(26,34)(27,39)(28,41)(29,40)(30,42)(44,45)(47,51)(48,53)(49,52)
(50,54)(55,71)(56,73)(57,72)(58,74)(59,67)(60,69)(61,68)(62,70)(63,75)(64,77)
(65,76)(66,78);;
s3 := ( 7,19)( 8,20)( 9,22)(10,21)(11,27)(12,28)(13,30)(14,29)(15,23)(16,24)
(17,26)(18,25)(31,35)(32,36)(33,38)(34,37)(41,42)(43,55)(44,56)(45,58)(46,57)
(47,63)(48,64)(49,66)(50,65)(51,59)(52,60)(53,62)(54,61)(67,71)(68,72)(69,74)
(70,73)(77,78);;
s4 := ( 7,46)( 8,45)( 9,44)(10,43)(11,50)(12,49)(13,48)(14,47)(15,54)(16,53)
(17,52)(18,51)(19,58)(20,57)(21,56)(22,55)(23,62)(24,61)(25,60)(26,59)(27,66)
(28,65)(29,64)(30,63)(31,70)(32,69)(33,68)(34,67)(35,74)(36,73)(37,72)(38,71)
(39,78)(40,77)(41,76)(42,75);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4*s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3, 
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(78)!(3,4)(5,6);
s1 := Sym(78)!(1,5)(2,3)(4,6);
s2 := Sym(78)!( 8, 9)(11,15)(12,17)(13,16)(14,18)(19,35)(20,37)(21,36)(22,38)
(23,31)(24,33)(25,32)(26,34)(27,39)(28,41)(29,40)(30,42)(44,45)(47,51)(48,53)
(49,52)(50,54)(55,71)(56,73)(57,72)(58,74)(59,67)(60,69)(61,68)(62,70)(63,75)
(64,77)(65,76)(66,78);
s3 := Sym(78)!( 7,19)( 8,20)( 9,22)(10,21)(11,27)(12,28)(13,30)(14,29)(15,23)
(16,24)(17,26)(18,25)(31,35)(32,36)(33,38)(34,37)(41,42)(43,55)(44,56)(45,58)
(46,57)(47,63)(48,64)(49,66)(50,65)(51,59)(52,60)(53,62)(54,61)(67,71)(68,72)
(69,74)(70,73)(77,78);
s4 := Sym(78)!( 7,46)( 8,45)( 9,44)(10,43)(11,50)(12,49)(13,48)(14,47)(15,54)
(16,53)(17,52)(18,51)(19,58)(20,57)(21,56)(22,55)(23,62)(24,61)(25,60)(26,59)
(27,66)(28,65)(29,64)(30,63)(31,70)(32,69)(33,68)(34,67)(35,74)(36,73)(37,72)
(38,71)(39,78)(40,77)(41,76)(42,75);
poly := sub<Sym(78)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4*s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope