Polytope of Type {3,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,6}*288
Also Known As : {{3,4},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(288,1028)
Rank : 4
Schlafli Type : {3,4,6}
Number of vertices, edges, etc : 6, 12, 24, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,4,6,2} of size 576
   {3,4,6,3} of size 864
   {3,4,6,4} of size 1152
   {3,4,6,3} of size 1152
   {3,4,6,4} of size 1152
   {3,4,6,6} of size 1728
   {3,4,6,6} of size 1728
   {3,4,6,6} of size 1728
Vertex Figure Of :
   {2,3,4,6} of size 576
   {3,3,4,6} of size 1152
   {4,3,4,6} of size 1152
   {6,3,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,4,2}*96
   4-fold quotients : {3,2,6}*72
   6-fold quotients : {3,4,2}*48
   8-fold quotients : {3,2,3}*36
   12-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,4,12}*576, {3,8,6}*576, {6,4,6}*576b
   3-fold covers : {3,4,18}*864, {9,4,6}*864, {3,12,6}*864a, {3,12,6}*864b
   4-fold covers : {3,8,12}*1152, {3,4,12}*1152, {3,8,6}*1152, {3,4,24}*1152, {12,4,6}*1152b, {6,4,12}*1152c, {6,4,6}*1152b, {12,4,6}*1152d, {6,8,6}*1152b, {6,8,6}*1152d
   5-fold covers : {15,4,6}*1440, {3,4,30}*1440
   6-fold covers : {3,4,36}*1728, {3,8,18}*1728, {9,4,12}*1728, {3,12,12}*1728a, {9,8,6}*1728, {3,24,6}*1728a, {6,4,18}*1728a, {18,4,6}*1728b, {6,12,6}*1728b, {3,24,6}*1728b, {3,12,12}*1728b, {6,12,6}*1728i, {6,12,6}*1728j, {6,12,6}*1728l
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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