Polytope of Type {2,12,6}

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

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

Permutation Representation (Magma) :

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

to this polytope