Polytope of Type {2,6,9,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,9,6}*1296
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 5
Schlafli Type : {2,6,9,6}
Number of vertices, edges, etc : 2, 6, 27, 27, 6
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,2,9,6}*432, {2,6,9,2}*432, {2,6,3,6}*432
9-fold quotients : {2,2,9,2}*144, {2,2,3,6}*144, {2,6,3,2}*144
27-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 6, 9)( 7,10)( 8,11)(15,18)(16,19)(17,20)(24,27)(25,28)(26,29)(33,36)
(34,37)(35,38)(42,45)(43,46)(44,47)(51,54)(52,55)(53,56)(60,63)(61,64)(62,65)
(69,72)(70,73)(71,74)(78,81)(79,82)(80,83);;
s2 := ( 3, 6)( 4, 8)( 5, 7)(10,11)(12,24)(13,26)(14,25)(15,21)(16,23)(17,22)
(18,27)(19,29)(20,28)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,65)(37,64)
(38,63)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,83)(46,82)(47,81)(48,71)
(49,70)(50,69)(51,68)(52,67)(53,66)(54,74)(55,73)(56,72);;
s3 := ( 3,39)( 4,41)( 5,40)( 6,45)( 7,47)( 8,46)( 9,42)(10,44)(11,43)(12,30)
(13,32)(14,31)(15,36)(16,38)(17,37)(18,33)(19,35)(20,34)(21,48)(22,50)(23,49)
(24,54)(25,56)(26,55)(27,51)(28,53)(29,52)(57,68)(58,67)(59,66)(60,74)(61,73)
(62,72)(63,71)(64,70)(65,69)(75,77)(78,83)(79,82)(80,81);;
s4 := (12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(39,48)
(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(66,75)(67,76)(68,77)
(69,78)(70,79)(71,80)(72,81)(73,82)(74,83);;
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*s1*s0*s1,
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*s1*s2*s1*s2*s3*s1*s2*s1*s2, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(83)!(1,2);
s1 := Sym(83)!( 6, 9)( 7,10)( 8,11)(15,18)(16,19)(17,20)(24,27)(25,28)(26,29)
(33,36)(34,37)(35,38)(42,45)(43,46)(44,47)(51,54)(52,55)(53,56)(60,63)(61,64)
(62,65)(69,72)(70,73)(71,74)(78,81)(79,82)(80,83);
s2 := Sym(83)!( 3, 6)( 4, 8)( 5, 7)(10,11)(12,24)(13,26)(14,25)(15,21)(16,23)
(17,22)(18,27)(19,29)(20,28)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,65)
(37,64)(38,63)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,83)(46,82)(47,81)
(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,74)(55,73)(56,72);
s3 := Sym(83)!( 3,39)( 4,41)( 5,40)( 6,45)( 7,47)( 8,46)( 9,42)(10,44)(11,43)
(12,30)(13,32)(14,31)(15,36)(16,38)(17,37)(18,33)(19,35)(20,34)(21,48)(22,50)
(23,49)(24,54)(25,56)(26,55)(27,51)(28,53)(29,52)(57,68)(58,67)(59,66)(60,74)
(61,73)(62,72)(63,71)(64,70)(65,69)(75,77)(78,83)(79,82)(80,81);
s4 := Sym(83)!(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)
(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(66,75)(67,76)
(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83);
poly := sub<Sym(83)|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*s1*s0*s1, 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*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope