Polytope of Type {3,6,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6,3}*1944b
if this polytope has a name.
Group : SmallGroup(1944,3578)
Rank : 5
Schlafli Type : {3,6,6,3}
Number of vertices, edges, etc : 9, 27, 54, 9, 3
Order of s₀s₁s₂s₃s₄ : 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 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 : {3,6,2,3}*648, {3,6,6,3}*648
   9-fold quotients : {3,2,6,3}*216, {3,6,2,3}*216
   27-fold quotients : {3,2,2,3}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)
(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);;
s1 := ( 1, 5)( 2, 6)( 3, 4)(10,14)(11,15)(12,13)(19,23)(20,24)(21,22)(28,32)
(29,33)(30,31)(37,41)(38,42)(39,40)(46,50)(47,51)(48,49)(55,59)(56,60)(57,58)
(64,68)(65,69)(66,67)(73,77)(74,78)(75,76);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,55)
(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,64)(38,66)(39,65)
(40,67)(41,69)(42,68)(43,70)(44,72)(45,71)(46,73)(47,75)(48,74)(49,76)(50,78)
(51,77)(52,79)(53,81)(54,80);;
s3 := ( 1,28)( 2,29)( 3,30)( 4,31)( 5,32)( 6,33)( 7,34)( 8,35)( 9,36)(10,46)
(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,37)(20,38)(21,39)
(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(64,73)(65,74)(66,75)(67,76)(68,77)
(69,78)(70,79)(71,80)(72,81);;
s4 := ( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)(28,64)
(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,55)(38,56)(39,57)
(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,73)(47,74)(48,75)(49,76)(50,77)
(51,78)(52,79)(53,80)(54,81);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4, s1*s2*s3*s2*s1*s2*s3*s2, 
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)
(60,63)(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);
s1 := Sym(81)!( 1, 5)( 2, 6)( 3, 4)(10,14)(11,15)(12,13)(19,23)(20,24)(21,22)
(28,32)(29,33)(30,31)(37,41)(38,42)(39,40)(46,50)(47,51)(48,49)(55,59)(56,60)
(57,58)(64,68)(65,69)(66,67)(73,77)(74,78)(75,76);
s2 := Sym(81)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)
(28,55)(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,64)(38,66)
(39,65)(40,67)(41,69)(42,68)(43,70)(44,72)(45,71)(46,73)(47,75)(48,74)(49,76)
(50,78)(51,77)(52,79)(53,81)(54,80);
s3 := Sym(81)!( 1,28)( 2,29)( 3,30)( 4,31)( 5,32)( 6,33)( 7,34)( 8,35)( 9,36)
(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,37)(20,38)
(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(64,73)(65,74)(66,75)(67,76)
(68,77)(69,78)(70,79)(71,80)(72,81);
s4 := Sym(81)!( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)
(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,55)(38,56)
(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,73)(47,74)(48,75)(49,76)
(50,77)(51,78)(52,79)(53,80)(54,81);
poly := sub<Sym(81)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4, 
s1*s2*s3*s2*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope