Polytope of Type {2,18,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,8}*576
if this polytope has a name.
Group : SmallGroup(576,1738)
Rank : 4
Schlafli Type : {2,18,8}
Number of vertices, edges, etc : 2, 18, 72, 8
Order of s₀s₁s₂s₃ : 72
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,18,8,2} of size 1152
Vertex Figure Of :
   {2,2,18,8} of size 1152
   {3,2,18,8} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,18,4}*288a
   3-fold quotients : {2,6,8}*192
   4-fold quotients : {2,18,2}*144
   6-fold quotients : {2,6,4}*96a
   8-fold quotients : {2,9,2}*72
   9-fold quotients : {2,2,8}*64
   12-fold quotients : {2,6,2}*48
   18-fold quotients : {2,2,4}*32
   24-fold quotients : {2,3,2}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,36,8}*1152a, {4,18,8}*1152a, {2,18,16}*1152
   3-fold covers : {2,54,8}*1728, {2,18,24}*1728a, {6,18,8}*1728a, {6,18,8}*1728b, {2,18,24}*1728b
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

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

to this polytope