Polytope of Type {18,4,4}

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

s0 := ( 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)(38,39)(40,45)(41,44)(42,43)(47,48)
(49,54)(50,53)(51,52)(56,57)(58,63)(59,62)(60,61)(65,66)(67,72)(68,71)
(69,70);;
s1 := ( 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)(37,49)(38,51)(39,50)(40,46)(41,48)
(42,47)(43,54)(44,53)(45,52)(55,67)(56,69)(57,68)(58,64)(59,66)(60,65)(61,72)
(62,71)(63,70);;
s2 := ( 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);;
s3 := (37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)
(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);;
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*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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(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)(38,39)(40,45)(41,44)(42,43)
(47,48)(49,54)(50,53)(51,52)(56,57)(58,63)(59,62)(60,61)(65,66)(67,72)(68,71)
(69,70);
s1 := 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)(37,49)(38,51)(39,50)(40,46)
(41,48)(42,47)(43,54)(44,53)(45,52)(55,67)(56,69)(57,68)(58,64)(59,66)(60,65)
(61,72)(62,71)(63,70);
s2 := 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);
s3 := Sym(72)!(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)
(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);
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*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
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