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

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

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

Permutation Representation (Magma) :

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

to this polytope