Polytope of Type {4,12,6}

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

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