Polytope of Type {6,2,4,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,4,18}*1728c
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 5
Schlafli Type : {6,2,4,18}
Number of vertices, edges, etc : 6, 6, 4, 36, 18
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 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) :
   2-fold quotients : {3,2,4,18}*864c, {6,2,4,9}*864
   3-fold quotients : {2,2,4,18}*576c, {6,2,4,6}*576b
   4-fold quotients : {3,2,4,9}*432
   6-fold quotients : {2,2,4,9}*288, {3,2,4,6}*288b, {6,2,4,3}*288
   9-fold quotients : {2,2,4,6}*192b
   12-fold quotients : {3,2,4,3}*144
   18-fold quotients : {2,2,4,3}*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 := ( 7,44)( 8,43)( 9,46)(10,45)(11,48)(12,47)(13,50)(14,49)(15,52)(16,51)
(17,54)(18,53)(19,56)(20,55)(21,58)(22,57)(23,60)(24,59)(25,62)(26,61)(27,64)
(28,63)(29,66)(30,65)(31,68)(32,67)(33,70)(34,69)(35,72)(36,71)(37,74)(38,73)
(39,76)(40,75)(41,78)(42,77);;
s3 := ( 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);;
s4 := ( 7,55)( 8,56)( 9,58)(10,57)(11,63)(12,64)(13,66)(14,65)(15,59)(16,60)
(17,62)(18,61)(19,43)(20,44)(21,46)(22,45)(23,51)(24,52)(25,54)(26,53)(27,47)
(28,48)(29,50)(30,49)(31,71)(32,72)(33,74)(34,73)(35,67)(36,68)(37,70)(38,69)
(39,75)(40,76)(41,78)(42,77);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s2*s4*s3*s4*s2*s3*s2 ];;
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)!( 7,44)( 8,43)( 9,46)(10,45)(11,48)(12,47)(13,50)(14,49)(15,52)
(16,51)(17,54)(18,53)(19,56)(20,55)(21,58)(22,57)(23,60)(24,59)(25,62)(26,61)
(27,64)(28,63)(29,66)(30,65)(31,68)(32,67)(33,70)(34,69)(35,72)(36,71)(37,74)
(38,73)(39,76)(40,75)(41,78)(42,77);
s3 := 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);
s4 := Sym(78)!( 7,55)( 8,56)( 9,58)(10,57)(11,63)(12,64)(13,66)(14,65)(15,59)
(16,60)(17,62)(18,61)(19,43)(20,44)(21,46)(22,45)(23,51)(24,52)(25,54)(26,53)
(27,47)(28,48)(29,50)(30,49)(31,71)(32,72)(33,74)(34,73)(35,67)(36,68)(37,70)
(38,69)(39,75)(40,76)(41,78)(42,77);
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s2*s4*s3*s4*s2*s3*s2 >;

to this polytope