Polytope of Type {6,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8,6}*576
Also Known As : {{6,8|2},{8,6|2}}. if this polytope has another name.
Group : SmallGroup(576,6606)
Rank : 4
Schlafli Type : {6,8,6}
Number of vertices, edges, etc : 6, 24, 24, 6
Order of s₀s₁s₂s₃ : 24
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,8,6,2} of size 1152
   {6,8,6,3} of size 1728
Vertex Figure Of :
   {2,6,8,6} of size 1152
   {3,6,8,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4,6}*288
   3-fold quotients : {2,8,6}*192, {6,8,2}*192
   4-fold quotients : {6,2,6}*144
   6-fold quotients : {2,4,6}*96a, {6,4,2}*96a
   8-fold quotients : {3,2,6}*72, {6,2,3}*72
   9-fold quotients : {2,8,2}*64
   12-fold quotients : {2,2,6}*48, {6,2,2}*48
   16-fold quotients : {3,2,3}*36
   18-fold quotients : {2,4,2}*32
   24-fold quotients : {2,2,3}*24, {3,2,2}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,8,12}*1152a, {12,8,6}*1152a, {6,16,6}*1152
   3-fold covers : {6,8,18}*1728, {18,8,6}*1728, {6,24,6}*1728a, {6,24,6}*1728b, {6,24,6}*1728f, {6,24,6}*1728g
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)(50,51)(53,54)(56,57)(59,60)(62,63)
(65,66)(68,69)(71,72);;
s1 := ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,29)(20,28)(21,30)(22,32)
(23,31)(24,33)(25,35)(26,34)(27,36)(37,56)(38,55)(39,57)(40,59)(41,58)(42,60)
(43,62)(44,61)(45,63)(46,65)(47,64)(48,66)(49,68)(50,67)(51,69)(52,71)(53,70)
(54,72);;
s2 := ( 1,37)( 2,38)( 3,39)( 4,43)( 5,44)( 6,45)( 7,40)( 8,41)( 9,42)(10,46)
(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,64)(20,65)(21,66)
(22,70)(23,71)(24,72)(25,67)(26,68)(27,69)(28,55)(29,56)(30,57)(31,61)(32,62)
(33,63)(34,58)(35,59)(36,60);;
s3 := ( 1, 4)( 2, 5)( 3, 6)(10,13)(11,14)(12,15)(19,22)(20,23)(21,24)(28,31)
(29,32)(30,33)(37,40)(38,41)(39,42)(46,49)(47,50)(48,51)(55,58)(56,59)(57,60)
(64,67)(65,68)(66,69);;
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*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
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