Polytope of Type {6,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,6}*1920b
if this polytope has a name.
Group : SmallGroup(1920,240973)
Rank : 4
Schlafli Type : {6,12,6}
Number of vertices, edges, etc : 10, 80, 80, 10
Order of s₀s₁s₂s₃ : 10
Order of s₀s₁s₂s₃s₂s₁ : 12
Special Properties :
   Universal
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,12,6}*960, {6,12,3}*960, {6,6,6}*960
   4-fold quotients : {3,12,3}*480, {3,6,6}*480, {6,3,6}*480, {6,6,3}*480
   8-fold quotients : {3,3,6}*240, {3,6,3}*240, {6,3,3}*240
   16-fold quotients : {3,3,3}*120
   120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 3,27)( 4,28)( 5,30)( 6,29)( 7,10)( 8, 9)(11,38)(12,37)(13,35)(14,36)
(15,16)(19,39)(20,40)(21,41)(22,42)(23,26)(24,25)(31,33)(32,34)(45,69)(46,70)
(47,72)(48,71)(49,52)(50,51)(53,80)(54,79)(55,77)(56,78)(57,58)(61,81)(62,82)
(63,83)(64,84)(65,68)(66,67)(73,75)(74,76);;
s1 := ( 1, 2)( 5, 6)( 7, 9)( 8,10)(11,14)(12,13)(15,27)(16,28)(17,30)(18,29)
(19,22)(20,21)(23,40)(24,39)(25,41)(26,42)(31,38)(32,37)(33,35)(34,36)(43,44)
(47,48)(49,51)(50,52)(53,56)(54,55)(57,69)(58,70)(59,72)(60,71)(61,64)(62,63)
(65,82)(66,81)(67,83)(68,84)(73,80)(74,79)(75,77)(76,78);;
s2 := ( 3,35)( 4,36)( 5,38)( 6,37)( 7,23)( 8,24)( 9,25)(10,26)(11,30)(12,29)
(13,27)(14,28)(15,16)(19,20)(31,34)(32,33)(39,40)(45,77)(46,78)(47,80)(48,79)
(49,65)(50,66)(51,67)(52,68)(53,72)(54,71)(55,69)(56,70)(57,58)(61,62)(73,76)
(74,75)(81,82);;
s3 := ( 1,43)( 2,44)( 3,45)( 4,46)( 5,48)( 6,47)( 7,52)( 8,51)( 9,50)(10,49)
(11,64)(12,63)(13,62)(14,61)(15,57)(16,58)(17,60)(18,59)(19,56)(20,55)(21,54)
(22,53)(23,75)(24,76)(25,74)(26,73)(27,69)(28,70)(29,72)(30,71)(31,68)(32,67)
(33,65)(34,66)(35,82)(36,81)(37,83)(38,84)(39,78)(40,77)(41,79)(42,80);;
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*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1, 
s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(84)!( 3,27)( 4,28)( 5,30)( 6,29)( 7,10)( 8, 9)(11,38)(12,37)(13,35)
(14,36)(15,16)(19,39)(20,40)(21,41)(22,42)(23,26)(24,25)(31,33)(32,34)(45,69)
(46,70)(47,72)(48,71)(49,52)(50,51)(53,80)(54,79)(55,77)(56,78)(57,58)(61,81)
(62,82)(63,83)(64,84)(65,68)(66,67)(73,75)(74,76);
s1 := Sym(84)!( 1, 2)( 5, 6)( 7, 9)( 8,10)(11,14)(12,13)(15,27)(16,28)(17,30)
(18,29)(19,22)(20,21)(23,40)(24,39)(25,41)(26,42)(31,38)(32,37)(33,35)(34,36)
(43,44)(47,48)(49,51)(50,52)(53,56)(54,55)(57,69)(58,70)(59,72)(60,71)(61,64)
(62,63)(65,82)(66,81)(67,83)(68,84)(73,80)(74,79)(75,77)(76,78);
s2 := Sym(84)!( 3,35)( 4,36)( 5,38)( 6,37)( 7,23)( 8,24)( 9,25)(10,26)(11,30)
(12,29)(13,27)(14,28)(15,16)(19,20)(31,34)(32,33)(39,40)(45,77)(46,78)(47,80)
(48,79)(49,65)(50,66)(51,67)(52,68)(53,72)(54,71)(55,69)(56,70)(57,58)(61,62)
(73,76)(74,75)(81,82);
s3 := Sym(84)!( 1,43)( 2,44)( 3,45)( 4,46)( 5,48)( 6,47)( 7,52)( 8,51)( 9,50)
(10,49)(11,64)(12,63)(13,62)(14,61)(15,57)(16,58)(17,60)(18,59)(19,56)(20,55)
(21,54)(22,53)(23,75)(24,76)(25,74)(26,73)(27,69)(28,70)(29,72)(30,71)(31,68)
(32,67)(33,65)(34,66)(35,82)(36,81)(37,83)(38,84)(39,78)(40,77)(41,79)(42,80);
poly := sub<Sym(84)|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*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1, 
s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope