Polytope of Type {6,4,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,8}*576
if this polytope has a name.
Group : SmallGroup(576,5339)
Rank : 4
Schlafli Type : {6,4,8}
Number of vertices, edges, etc : 9, 18, 24, 8
Order of s₀s₁s₂s₃ : 8
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,4,8,2} of size 1152
Vertex Figure Of :
   {2,6,4,8} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4,4}*288
   4-fold quotients : {6,4,2}*144
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,4,16}*1152, {6,4,8}*1152a
   3-fold covers : {6,4,8}*1728, {6,12,8}*1728a, {6,12,8}*1728b, {6,4,24}*1728, {6,12,8}*1728c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope