Polytope of Type {2,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,12}*1296
if this polytope has a name.
Group : SmallGroup(1296,2977)
Rank : 4
Schlafli Type : {2,6,12}
Number of vertices, edges, etc : 2, 27, 162, 54
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {2,6,4}*432, {2,6,12}*432a, {2,6,12}*432b, {2,6,12}*432c
   9-fold quotients : {2,6,4}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope