Polytope of Type {12,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,6}*576
Also Known As : {{12,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(576,6139)
Rank : 4
Schlafli Type : {12,4,6}
Number of vertices, edges, etc : 12, 24, 12, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,4,6,2} of size 1152
   {12,4,6,3} of size 1728
Vertex Figure Of :
   {2,12,4,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,2,6}*288, {6,4,6}*288
   3-fold quotients : {12,4,2}*192a, {4,4,6}*192
   4-fold quotients : {12,2,3}*144, {6,2,6}*144
   6-fold quotients : {12,2,2}*96, {2,4,6}*96a, {4,2,6}*96, {6,4,2}*96a
   8-fold quotients : {3,2,6}*72, {6,2,3}*72
   9-fold quotients : {4,4,2}*64
   12-fold quotients : {4,2,3}*48, {2,2,6}*48, {6,2,2}*48
   16-fold quotients : {3,2,3}*36
   18-fold quotients : {2,4,2}*32, {4,2,2}*32
   24-fold quotients : {2,2,3}*24, {3,2,2}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,4,12}*1152, {12,8,6}*1152a, {24,4,6}*1152a, {12,8,6}*1152b, {24,4,6}*1152b, {12,4,6}*1152a
   3-fold covers : {12,4,18}*1728, {36,4,6}*1728, {12,12,6}*1728a, {12,12,6}*1728b, {12,12,6}*1728f, {12,12,6}*1728g
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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