Polytope of Type {4,4,6}

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

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

Permutation Representation (Magma) :

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

References : None.
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