Polytope of Type {2,3,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,3}*1944
if this polytope has a name.
Group : SmallGroup(1944,941)
Rank : 5
Schlafli Type : {2,3,6,3}
Number of vertices, edges, etc : 2, 9, 81, 81, 9
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {2,3,6,3}*648a, {2,3,6,3}*648b
   9-fold quotients : {2,3,6,3}*216
   27-fold quotients : {2,3,2,3}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 5)( 7, 8)(10,11)(12,24)(13,26)(14,25)(15,27)(16,29)(17,28)(18,21)
(19,23)(20,22)(30,57)(31,59)(32,58)(33,60)(34,62)(35,61)(36,63)(37,65)(38,64)
(39,78)(40,80)(41,79)(42,81)(43,83)(44,82)(45,75)(46,77)(47,76)(48,72)(49,74)
(50,73)(51,66)(52,68)(53,67)(54,69)(55,71)(56,70);;
s2 := ( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,38)(11,37)(12,48)
(13,50)(14,49)(15,51)(16,53)(17,52)(18,54)(19,56)(20,55)(21,39)(22,41)(23,40)
(24,42)(25,44)(26,43)(27,45)(28,47)(29,46)(58,59)(61,62)(64,65)(66,75)(67,77)
(68,76)(69,78)(70,80)(71,79)(72,81)(73,83)(74,82);;
s3 := ( 6, 9)( 7,10)( 8,11)(12,21)(13,22)(14,23)(15,27)(16,28)(17,29)(18,24)
(19,25)(20,26)(30,39)(31,40)(32,41)(33,45)(34,46)(35,47)(36,42)(37,43)(38,44)
(51,54)(52,55)(53,56)(57,78)(58,79)(59,80)(60,75)(61,76)(62,77)(63,81)(64,82)
(65,83)(66,69)(67,70)(68,71);;
s4 := ( 6, 9)( 7,10)( 8,11)(12,22)(13,23)(14,21)(15,28)(16,29)(17,27)(18,25)
(19,26)(20,24)(33,36)(34,37)(35,38)(39,49)(40,50)(41,48)(42,55)(43,56)(44,54)
(45,52)(46,53)(47,51)(60,63)(61,64)(62,65)(66,76)(67,77)(68,75)(69,82)(70,83)
(71,81)(72,79)(73,80)(74,78);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s1*s2*s1*s2, s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3, 
s3*s1*s4*s3*s2*s3*s2*s3*s1*s2*s4*s3*s4*s1*s2*s3*s2*s3*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(83)!(1,2);
s1 := Sym(83)!( 4, 5)( 7, 8)(10,11)(12,24)(13,26)(14,25)(15,27)(16,29)(17,28)
(18,21)(19,23)(20,22)(30,57)(31,59)(32,58)(33,60)(34,62)(35,61)(36,63)(37,65)
(38,64)(39,78)(40,80)(41,79)(42,81)(43,83)(44,82)(45,75)(46,77)(47,76)(48,72)
(49,74)(50,73)(51,66)(52,68)(53,67)(54,69)(55,71)(56,70);
s2 := Sym(83)!( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,38)(11,37)
(12,48)(13,50)(14,49)(15,51)(16,53)(17,52)(18,54)(19,56)(20,55)(21,39)(22,41)
(23,40)(24,42)(25,44)(26,43)(27,45)(28,47)(29,46)(58,59)(61,62)(64,65)(66,75)
(67,77)(68,76)(69,78)(70,80)(71,79)(72,81)(73,83)(74,82);
s3 := Sym(83)!( 6, 9)( 7,10)( 8,11)(12,21)(13,22)(14,23)(15,27)(16,28)(17,29)
(18,24)(19,25)(20,26)(30,39)(31,40)(32,41)(33,45)(34,46)(35,47)(36,42)(37,43)
(38,44)(51,54)(52,55)(53,56)(57,78)(58,79)(59,80)(60,75)(61,76)(62,77)(63,81)
(64,82)(65,83)(66,69)(67,70)(68,71);
s4 := Sym(83)!( 6, 9)( 7,10)( 8,11)(12,22)(13,23)(14,21)(15,28)(16,29)(17,27)
(18,25)(19,26)(20,24)(33,36)(34,37)(35,38)(39,49)(40,50)(41,48)(42,55)(43,56)
(44,54)(45,52)(46,53)(47,51)(60,63)(61,64)(62,65)(66,76)(67,77)(68,75)(69,82)
(70,83)(71,81)(72,79)(73,80)(74,78);
poly := sub<Sym(83)|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*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2, 
s3*s4*s3*s4*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3, 
s3*s1*s4*s3*s2*s3*s2*s3*s1*s2*s4*s3*s4*s1*s2*s3*s2*s3*s1*s2 >;

to this polytope