Polytope of Type {3,6,4}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope