Polytope of Type {6,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,4}*576a
Also Known As : {{6,12|2},{12,4|2}}. if this polytope has another name.
Group : SmallGroup(576,6139)
Rank : 4
Schlafli Type : {6,12,4}
Number of vertices, edges, etc : 6, 36, 24, 4
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,12,4,2} of size 1152
Vertex Figure Of :
   {2,6,12,4} of size 1152
   {3,6,12,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,12,2}*288a, {6,6,4}*288a
   3-fold quotients : {2,12,4}*192a, {6,4,4}*192
   4-fold quotients : {6,6,2}*144a
   6-fold quotients : {2,12,2}*96, {2,6,4}*96a, {6,2,4}*96, {6,4,2}*96a
   9-fold quotients : {2,4,4}*64
   12-fold quotients : {3,2,4}*48, {2,6,2}*48, {6,2,2}*48
   18-fold quotients : {2,2,4}*32, {2,4,2}*32
   24-fold quotients : {2,3,2}*24, {3,2,2}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12,4}*1152b, {6,12,8}*1152b, {6,24,4}*1152c, {6,12,8}*1152e, {6,24,4}*1152f, {6,12,4}*1152b
   3-fold covers : {18,12,4}*1728a, {6,36,4}*1728a, {6,12,4}*1728b, {6,12,12}*1728b, {6,12,12}*1728e, {6,12,4}*1728j
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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