Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1296s
if this polytope has a name.
Group : SmallGroup(1296,3538)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 18, 54, 54, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6,6}*432b, {6,6,6}*432d, {6,6,6}*432g, {6,6,2}*432d
   6-fold quotients : {3,6,6}*216b
   9-fold quotients : {2,6,6}*144a, {6,2,6}*144, {6,6,2}*144a, {6,6,2}*144b, {6,6,2}*144c
   18-fold quotients : {3,2,6}*72, {3,6,2}*72, {6,2,3}*72, {6,3,2}*72
   27-fold quotients : {2,2,6}*48, {2,6,2}*48, {6,2,2}*48
   36-fold quotients : {3,2,3}*36
   54-fold quotients : {2,2,3}*24, {2,3,2}*24, {3,2,2}*24
   81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope