Polytope of Type {4,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,3}*144
if this polytope has a name.
Group : SmallGroup(144,154)
Rank : 4
Schlafli Type : {4,6,3}
Number of vertices, edges, etc : 4, 12, 9, 3
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,6,3,2} of size 288
   {4,6,3,4} of size 576
   {4,6,3,6} of size 864
   {4,6,3,4} of size 1152
Vertex Figure Of :
   {2,4,6,3} of size 288
   {4,4,6,3} of size 576
   {6,4,6,3} of size 864
   {3,4,6,3} of size 864
   {8,4,6,3} of size 1152
   {8,4,6,3} of size 1152
   {4,4,6,3} of size 1152
   {6,4,6,3} of size 1296
   {10,4,6,3} of size 1440
   {12,4,6,3} of size 1728
   {6,4,6,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,6,3}*72
   3-fold quotients : {4,2,3}*48
   6-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,6,3}*288, {4,6,6}*288c
   3-fold covers : {4,6,9}*432, {4,6,3}*432a, {12,6,3}*432b
   4-fold covers : {16,6,3}*576, {4,6,12}*576b, {8,6,6}*576c, {4,12,6}*576c, {4,6,3}*576a, {4,12,3}*576
   5-fold covers : {20,6,3}*720, {4,6,15}*720
   6-fold covers : {8,6,9}*864, {8,6,3}*864a, {4,6,18}*864b, {4,6,6}*864c, {24,6,3}*864b, {12,6,6}*864e, {4,6,6}*864h
   7-fold covers : {28,6,3}*1008, {4,6,21}*1008
   8-fold covers : {32,6,3}*1152, {4,12,12}*1152c, {4,24,6}*1152a, {8,12,6}*1152c, {4,24,6}*1152d, {8,12,6}*1152f, {4,12,6}*1152c, {8,6,12}*1152c, {4,6,24}*1152c, {16,6,6}*1152c, {8,6,3}*1152, {8,12,3}*1152, {4,12,3}*1152b, {4,24,3}*1152, {4,6,6}*1152f, {4,12,6}*1152j
   9-fold covers : {4,18,9}*1296, {4,6,9}*1296a, {4,6,27}*1296, {4,6,9}*1296b, {4,6,9}*1296c, {4,6,9}*1296d, {4,6,3}*1296a, {4,18,3}*1296, {36,6,3}*1296b, {12,6,9}*1296b, {12,6,3}*1296c, {12,6,3}*1296d, {12,6,3}*1296e, {4,6,3}*1296b
   10-fold covers : {40,6,3}*1440, {8,6,15}*1440, {20,6,6}*1440c, {4,30,6}*1440a, {4,6,30}*1440c
   11-fold covers : {44,6,3}*1584, {4,6,33}*1584
   12-fold covers : {16,6,9}*1728, {16,6,3}*1728a, {4,6,36}*1728b, {4,6,12}*1728b, {8,6,18}*1728b, {8,6,6}*1728c, {4,12,18}*1728b, {4,12,6}*1728c, {48,6,3}*1728b, {4,6,9}*1728a, {4,12,9}*1728, {4,6,3}*1728a, {4,12,3}*1728a, {24,6,6}*1728e, {12,6,12}*1728e, {8,6,6}*1728e, {4,12,6}*1728j, {12,12,6}*1728g, {4,6,12}*1728h, {12,6,3}*1728, {12,12,3}*1728b
   13-fold covers : {52,6,3}*1872, {4,6,39}*1872
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope