Polytope of Type {6,2,4,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,4,6,3}*1728
if this polytope has a name.
Group : SmallGroup(1728,47409)
Rank : 6
Schlafli Type : {6,2,4,6,3}
Number of vertices, edges, etc : 6, 6, 4, 12, 9, 3
Order of s₀s₁s₂s₃s₄s₅ : 12
Order of s₀s₁s₂s₃s₄s₅s₄s₃s₂s₁ : 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 : {3,2,4,6,3}*864, {6,2,2,6,3}*864
   3-fold quotients : {6,2,4,2,3}*576, {2,2,4,6,3}*576
   4-fold quotients : {3,2,2,6,3}*432
   6-fold quotients : {3,2,4,2,3}*288, {2,2,2,6,3}*288, {6,2,2,2,3}*288
   9-fold quotients : {2,2,4,2,3}*192
   12-fold quotients : {3,2,2,2,3}*144
   18-fold quotients : {2,2,2,2,3}*96
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 := ( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)
(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(43,61)(44,62)(45,63)
(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)
(57,75)(58,76)(59,77)(60,78);;
s3 := ( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(25,34)(26,36)(27,35)(28,37)
(29,39)(30,38)(31,40)(32,42)(33,41)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)
(61,70)(62,72)(63,71)(64,73)(65,75)(66,74)(67,76)(68,78)(69,77);;
s4 := ( 7,44)( 8,43)( 9,45)(10,50)(11,49)(12,51)(13,47)(14,46)(15,48)(16,53)
(17,52)(18,54)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)(27,63)
(28,68)(29,67)(30,69)(31,65)(32,64)(33,66)(34,71)(35,70)(36,72)(37,77)(38,76)
(39,78)(40,74)(41,73)(42,75);;
s5 := ( 7,46)( 8,48)( 9,47)(10,43)(11,45)(12,44)(13,49)(14,51)(15,50)(16,55)
(17,57)(18,56)(19,52)(20,54)(21,53)(22,58)(23,60)(24,59)(25,64)(26,66)(27,65)
(28,61)(29,63)(30,62)(31,67)(32,69)(33,68)(34,73)(35,75)(36,74)(37,70)(38,72)
(39,71)(40,76)(41,78)(42,77);;
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*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, s0*s5*s0*s5, s1*s5*s1*s5, 
s2*s5*s2*s5, s3*s5*s3*s5, s4*s5*s4*s5*s4*s5, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
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)!( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)
(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(43,61)(44,62)
(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)
(56,74)(57,75)(58,76)(59,77)(60,78);
s3 := Sym(78)!( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(25,34)(26,36)(27,35)
(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(44,45)(47,48)(50,51)(53,54)(56,57)
(59,60)(61,70)(62,72)(63,71)(64,73)(65,75)(66,74)(67,76)(68,78)(69,77);
s4 := Sym(78)!( 7,44)( 8,43)( 9,45)(10,50)(11,49)(12,51)(13,47)(14,46)(15,48)
(16,53)(17,52)(18,54)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)
(27,63)(28,68)(29,67)(30,69)(31,65)(32,64)(33,66)(34,71)(35,70)(36,72)(37,77)
(38,76)(39,78)(40,74)(41,73)(42,75);
s5 := Sym(78)!( 7,46)( 8,48)( 9,47)(10,43)(11,45)(12,44)(13,49)(14,51)(15,50)
(16,55)(17,57)(18,56)(19,52)(20,54)(21,53)(22,58)(23,60)(24,59)(25,64)(26,66)
(27,65)(28,61)(29,63)(30,62)(31,67)(32,69)(33,68)(34,73)(35,75)(36,74)(37,70)
(38,72)(39,71)(40,76)(41,78)(42,77);
poly := sub<Sym(78)|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*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, s0*s5*s0*s5, 
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5, 
s4*s5*s4*s5*s4*s5, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

to this polytope