Polytope of Type {3,6,18}

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

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