Polytope of Type {4,4,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,18}*576
Also Known As : {{4,4|2},{4,18|2}}. if this polytope has another name.
Group : SmallGroup(576,1572)
Rank : 4
Schlafli Type : {4,4,18}
Number of vertices, edges, etc : 4, 8, 36, 18
Order of s₀s₁s₂s₃ : 36
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,4,18,2} of size 1152
Vertex Figure Of :
   {2,4,4,18} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,18}*288a, {4,2,18}*288
   3-fold quotients : {4,4,6}*192
   4-fold quotients : {4,2,9}*144, {2,2,18}*144
   6-fold quotients : {2,4,6}*96a, {4,2,6}*96
   8-fold quotients : {2,2,9}*72
   9-fold quotients : {4,4,2}*64
   12-fold quotients : {4,2,3}*48, {2,2,6}*48
   18-fold quotients : {2,4,2}*32, {4,2,2}*32
   24-fold quotients : {2,2,3}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4,36}*1152, {4,8,18}*1152a, {8,4,18}*1152a, {4,8,18}*1152b, {8,4,18}*1152b, {4,4,18}*1152a
   3-fold covers : {4,4,54}*1728, {4,12,18}*1728a, {12,4,18}*1728, {4,12,18}*1728b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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