Polytope of Type {6,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,4}*192
Also Known As : {{6,4|2},{4,4|2}}. if this polytope has another name.
Group : SmallGroup(192,1147)
Rank : 4
Schlafli Type : {6,4,4}
Number of vertices, edges, etc : 6, 12, 8, 4
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,4,4,2} of size 384
   {6,4,4,4} of size 768
   {6,4,4,6} of size 1152
   {6,4,4,3} of size 1152
   {6,4,4,6} of size 1728
   {6,4,4,10} of size 1920
Vertex Figure Of :
   {2,6,4,4} of size 384
   {3,6,4,4} of size 576
   {4,6,4,4} of size 768
   {3,6,4,4} of size 768
   {4,6,4,4} of size 768
   {6,6,4,4} of size 1152
   {6,6,4,4} of size 1152
   {6,6,4,4} of size 1152
   {9,6,4,4} of size 1728
   {3,6,4,4} of size 1728
   {10,6,4,4} of size 1920
   {5,6,4,4} of size 1920
   {5,6,4,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,2,4}*96, {6,4,2}*96a
   3-fold quotients : {2,4,4}*64
   4-fold quotients : {3,2,4}*48, {6,2,2}*48
   6-fold quotients : {2,2,4}*32, {2,4,2}*32
   8-fold quotients : {3,2,2}*24
   12-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,4,4}*384, {6,4,8}*384a, {6,8,4}*384a, {6,4,8}*384b, {6,8,4}*384b, {6,4,4}*384a
   3-fold covers : {18,4,4}*576, {6,4,12}*576, {6,12,4}*576a, {6,12,4}*576c
   4-fold covers : {6,4,8}*768a, {6,8,4}*768a, {6,8,8}*768a, {6,8,8}*768b, {6,8,8}*768c, {6,8,8}*768d, {12,4,8}*768a, {24,4,4}*768a, {12,4,8}*768b, {24,4,4}*768b, {12,8,4}*768a, {12,4,4}*768a, {12,4,4}*768b, {12,8,4}*768b, {12,8,4}*768c, {12,8,4}*768d, {6,4,16}*768a, {6,16,4}*768a, {6,4,16}*768b, {6,16,4}*768b, {6,4,4}*768a, {6,4,8}*768b, {6,8,4}*768b, {6,4,4}*768e
   5-fold covers : {6,4,20}*960, {6,20,4}*960, {30,4,4}*960
   6-fold covers : {36,4,4}*1152, {12,12,4}*1152b, {12,12,4}*1152c, {12,4,12}*1152, {18,4,8}*1152a, {18,8,4}*1152a, {6,8,12}*1152a, {6,12,8}*1152b, {6,12,8}*1152c, {6,24,4}*1152a, {6,4,24}*1152a, {6,24,4}*1152c, {18,4,8}*1152b, {18,8,4}*1152b, {6,8,12}*1152b, {6,12,8}*1152e, {6,12,8}*1152f, {6,24,4}*1152d, {6,4,24}*1152b, {6,24,4}*1152f, {18,4,4}*1152a, {6,4,12}*1152a, {6,12,4}*1152b, {6,12,4}*1152c
   7-fold covers : {6,4,28}*1344, {6,28,4}*1344, {42,4,4}*1344
   9-fold covers : {54,4,4}*1728, {18,4,12}*1728, {18,12,4}*1728a, {6,4,36}*1728, {6,36,4}*1728a, {6,12,4}*1728b, {6,12,12}*1728a, {18,12,4}*1728b, {6,12,4}*1728c, {6,12,12}*1728b, {6,12,12}*1728c, {6,12,12}*1728e, {6,12,4}*1728j, {6,12,12}*1728g, {6,4,4}*1728b, {6,4,4}*1728c, {6,4,12}*1728b, {6,12,4}*1728n, {6,12,4}*1728p
   10-fold covers : {60,4,4}*1920, {12,20,4}*1920, {12,4,20}*1920, {30,4,8}*1920a, {30,8,4}*1920a, {6,8,20}*1920a, {6,20,8}*1920a, {6,4,40}*1920a, {6,40,4}*1920a, {30,4,8}*1920b, {30,8,4}*1920b, {6,8,20}*1920b, {6,20,8}*1920b, {6,4,40}*1920b, {6,40,4}*1920b, {30,4,4}*1920a, {6,4,20}*1920a, {6,20,4}*1920a
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)
(32,33)(35,36)(38,39)(41,42)(44,45)(47,48);;
s1 := ( 1,14)( 2,13)( 3,15)( 4,17)( 5,16)( 6,18)( 7,20)( 8,19)( 9,21)(10,23)
(11,22)(12,24)(25,38)(26,37)(27,39)(28,41)(29,40)(30,42)(31,44)(32,43)(33,45)
(34,47)(35,46)(36,48);;
s2 := (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,28)(26,29)(27,30)(31,34)
(32,35)(33,36)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);;
s3 := ( 1,25)( 2,26)( 3,27)( 4,28)( 5,29)( 6,30)( 7,31)( 8,32)( 9,33)(10,34)
(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)
(22,46)(23,47)(24,48);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(48)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)
(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48);
s1 := Sym(48)!( 1,14)( 2,13)( 3,15)( 4,17)( 5,16)( 6,18)( 7,20)( 8,19)( 9,21)
(10,23)(11,22)(12,24)(25,38)(26,37)(27,39)(28,41)(29,40)(30,42)(31,44)(32,43)
(33,45)(34,47)(35,46)(36,48);
s2 := Sym(48)!(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,28)(26,29)(27,30)
(31,34)(32,35)(33,36)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);
s3 := Sym(48)!( 1,25)( 2,26)( 3,27)( 4,28)( 5,29)( 6,30)( 7,31)( 8,32)( 9,33)
(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)
(21,45)(22,46)(23,47)(24,48);
poly := sub<Sym(48)|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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
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