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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,4,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,47887)
Rank : 5
Schlafli Type : {6,2,4,6}
Number of vertices, edges, etc : 6, 6, 12, 36, 18
Order of s₀s₁s₂s₃s₄ : 12
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 : {3,2,4,6}*864, {6,2,4,6}*864
   3-fold quotients : {2,2,4,6}*576
   4-fold quotients : {3,2,4,6}*432
   6-fold quotients : {2,2,4,6}*288
   9-fold quotients : {6,2,4,2}*192
   18-fold quotients : {3,2,4,2}*96, {6,2,2,2}*96
   27-fold quotients : {2,2,4,2}*64
   36-fold quotients : {3,2,2,2}*48
   54-fold quotients : {2,2,2,2}*32
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 := (10,13)(11,14)(12,15)(19,22)(20,23)(21,24)(28,31)(29,32)(30,33)(37,40)
(38,41)(39,42)(43,52)(44,53)(45,54)(46,58)(47,59)(48,60)(49,55)(50,56)(51,57)
(61,70)(62,71)(63,72)(64,76)(65,77)(66,78)(67,73)(68,74)(69,75);;
s3 := ( 7,43)( 8,46)( 9,49)(10,44)(11,47)(12,50)(13,45)(14,48)(15,51)(16,52)
(17,55)(18,58)(19,53)(20,56)(21,59)(22,54)(23,57)(24,60)(25,61)(26,64)(27,67)
(28,62)(29,65)(30,68)(31,63)(32,66)(33,69)(34,70)(35,73)(36,76)(37,71)(38,74)
(39,77)(40,72)(41,75)(42,78);;
s4 := ( 7,35)( 8,34)( 9,36)(10,41)(11,40)(12,42)(13,38)(14,37)(15,39)(16,26)
(17,25)(18,27)(19,32)(20,31)(21,33)(22,29)(23,28)(24,30)(43,71)(44,70)(45,72)
(46,77)(47,76)(48,78)(49,74)(50,73)(51,75)(52,62)(53,61)(54,63)(55,68)(56,67)
(57,69)(58,65)(59,64)(60,66);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
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)!(10,13)(11,14)(12,15)(19,22)(20,23)(21,24)(28,31)(29,32)(30,33)
(37,40)(38,41)(39,42)(43,52)(44,53)(45,54)(46,58)(47,59)(48,60)(49,55)(50,56)
(51,57)(61,70)(62,71)(63,72)(64,76)(65,77)(66,78)(67,73)(68,74)(69,75);
s3 := Sym(78)!( 7,43)( 8,46)( 9,49)(10,44)(11,47)(12,50)(13,45)(14,48)(15,51)
(16,52)(17,55)(18,58)(19,53)(20,56)(21,59)(22,54)(23,57)(24,60)(25,61)(26,64)
(27,67)(28,62)(29,65)(30,68)(31,63)(32,66)(33,69)(34,70)(35,73)(36,76)(37,71)
(38,74)(39,77)(40,72)(41,75)(42,78);
s4 := Sym(78)!( 7,35)( 8,34)( 9,36)(10,41)(11,40)(12,42)(13,38)(14,37)(15,39)
(16,26)(17,25)(18,27)(19,32)(20,31)(21,33)(22,29)(23,28)(24,30)(43,71)(44,70)
(45,72)(46,77)(47,76)(48,78)(49,74)(50,73)(51,75)(52,62)(53,61)(54,63)(55,68)
(56,67)(57,69)(58,65)(59,64)(60,66);
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

to this polytope