Polytope of Type {18,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,3}*648b
if this polytope has a name.
Group : SmallGroup(648,554)
Rank : 4
Schlafli Type : {18,6,3}
Number of vertices, edges, etc : 18, 54, 9, 3
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {18,6,3,2} of size 1296
Vertex Figure Of :
   {2,18,6,3} of size 1296
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,2,3}*216, {6,6,3}*216b
   6-fold quotients : {9,2,3}*108
   9-fold quotients : {2,6,3}*72, {6,2,3}*72
   18-fold quotients : {3,2,3}*36
   27-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,6,3}*1296b, {18,6,6}*1296c
   3-fold covers : {18,6,9}*1944b, {18,6,3}*1944c, {18,6,3}*1944d, {18,6,3}*1944e, {54,6,3}*1944b
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,57)
(29,56)(30,55)(31,60)(32,59)(33,58)(34,63)(35,62)(36,61)(37,66)(38,65)(39,64)
(40,69)(41,68)(42,67)(43,72)(44,71)(45,70)(46,75)(47,74)(48,73)(49,78)(50,77)
(51,76)(52,81)(53,80)(54,79);;
s1 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,37)
(11,39)(12,38)(13,43)(14,45)(15,44)(16,40)(17,42)(18,41)(19,46)(20,48)(21,47)
(22,52)(23,54)(24,53)(25,49)(26,51)(27,50)(55,57)(58,63)(59,62)(60,61)(64,66)
(67,72)(68,71)(69,70)(73,75)(76,81)(77,80)(78,79);;
s2 := ( 1, 4)( 2, 5)( 3, 6)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)(16,25)
(17,26)(18,27)(28,31)(29,32)(30,33)(37,49)(38,50)(39,51)(40,46)(41,47)(42,48)
(43,52)(44,53)(45,54)(55,58)(56,59)(57,60)(64,76)(65,77)(66,78)(67,73)(68,74)
(69,75)(70,79)(71,80)(72,81);;
s3 := ( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)(22,25)
(23,26)(24,27)(28,37)(29,38)(30,39)(31,43)(32,44)(33,45)(34,40)(35,41)(36,42)
(49,52)(50,53)(51,54)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)
(63,69)(76,79)(77,80)(78,81);;
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, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
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(81)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)
(28,57)(29,56)(30,55)(31,60)(32,59)(33,58)(34,63)(35,62)(36,61)(37,66)(38,65)
(39,64)(40,69)(41,68)(42,67)(43,72)(44,71)(45,70)(46,75)(47,74)(48,73)(49,78)
(50,77)(51,76)(52,81)(53,80)(54,79);
s1 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,37)(11,39)(12,38)(13,43)(14,45)(15,44)(16,40)(17,42)(18,41)(19,46)(20,48)
(21,47)(22,52)(23,54)(24,53)(25,49)(26,51)(27,50)(55,57)(58,63)(59,62)(60,61)
(64,66)(67,72)(68,71)(69,70)(73,75)(76,81)(77,80)(78,79);
s2 := Sym(81)!( 1, 4)( 2, 5)( 3, 6)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)
(16,25)(17,26)(18,27)(28,31)(29,32)(30,33)(37,49)(38,50)(39,51)(40,46)(41,47)
(42,48)(43,52)(44,53)(45,54)(55,58)(56,59)(57,60)(64,76)(65,77)(66,78)(67,73)
(68,74)(69,75)(70,79)(71,80)(72,81);
s3 := Sym(81)!( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)
(22,25)(23,26)(24,27)(28,37)(29,38)(30,39)(31,43)(32,44)(33,45)(34,40)(35,41)
(36,42)(49,52)(50,53)(51,54)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)
(62,68)(63,69)(76,79)(77,80)(78,81);
poly := sub<Sym(81)|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, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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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