Polytope of Type {3,6,8,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,8,2}*576
if this polytope has a name.
Group : SmallGroup(576,6980)
Rank : 5
Schlafli Type : {3,6,8,2}
Number of vertices, edges, etc : 3, 9, 24, 8, 2
Order of s₀s₁s₂s₃s₄ : 24
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,8,2,2} of size 1152
   {3,6,8,2,3} of size 1728
Vertex Figure Of :
   {2,3,6,8,2} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,4,2}*288
   3-fold quotients : {3,2,8,2}*192
   4-fold quotients : {3,6,2,2}*144
   6-fold quotients : {3,2,4,2}*96
   12-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,8,4}*1152a, {3,6,16,2}*1152, {6,6,8,2}*1152c
   3-fold covers : {9,6,8,2}*1728, {3,6,8,2}*1728a, {3,6,24,2}*1728b, {3,6,8,6}*1728
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, 5)( 2, 4)( 3, 6)( 7, 8)(10,14)(11,13)(12,15)(16,17)(19,23)(20,22)
(21,24)(25,26)(28,32)(29,31)(30,33)(34,35)(37,41)(38,40)(39,42)(43,44)(46,50)
(47,49)(48,51)(52,53)(55,59)(56,58)(57,60)(61,62)(64,68)(65,67)(66,69)
(70,71);;
s2 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,34)
(23,35)(24,36)(25,31)(26,32)(27,33)(37,55)(38,56)(39,57)(40,61)(41,62)(42,63)
(43,58)(44,59)(45,60)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)
(54,69);;
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);;
s4 := (73,74);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*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(74)!( 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(74)!( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,14)(11,13)(12,15)(16,17)(19,23)
(20,22)(21,24)(25,26)(28,32)(29,31)(30,33)(34,35)(37,41)(38,40)(39,42)(43,44)
(46,50)(47,49)(48,51)(52,53)(55,59)(56,58)(57,60)(61,62)(64,68)(65,67)(66,69)
(70,71);
s2 := Sym(74)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)
(22,34)(23,35)(24,36)(25,31)(26,32)(27,33)(37,55)(38,56)(39,57)(40,61)(41,62)
(42,63)(43,58)(44,59)(45,60)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)
(53,68)(54,69);
s3 := Sym(74)!( 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);
s4 := Sym(74)!(73,74);
poly := sub<Sym(74)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope