Polytope of Type {12,6,2,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,2,3}*1728a
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {12,6,2,3}
Number of vertices, edges, etc : 24, 72, 12, 3, 3
Order of s₀s₁s₂s₃s₄ : 6
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) :
   2-fold quotients : {12,6,2,3}*864d
   3-fold quotients : {4,6,2,3}*576
   4-fold quotients : {6,6,2,3}*432a
   6-fold quotients : {4,3,2,3}*288, {4,6,2,3}*288b, {4,6,2,3}*288c
   12-fold quotients : {4,3,2,3}*144, {2,6,2,3}*144, {6,2,2,3}*144
   24-fold quotients : {2,3,2,3}*72, {3,2,2,3}*72
   36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope