Polytope of Type {2,6,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6,9}*1296b
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 5
Schlafli Type : {2,6,6,9}
Number of vertices, edges, etc : 2, 6, 18, 27, 9
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 : {2,2,6,9}*432, {2,6,2,9}*432, {2,6,6,3}*432b
6-fold quotients : {2,3,2,9}*216
9-fold quotients : {2,2,2,9}*144, {2,2,6,3}*144, {2,6,2,3}*144
18-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,2,2,3}*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, 7)( 5, 8)(12,24)(13,25)(14,26)(15,21)(16,22)(17,23)(18,27)
(19,28)(20,29)(30,33)(31,34)(32,35)(39,51)(40,52)(41,53)(42,48)(43,49)(44,50)
(45,54)(46,55)(47,56)(57,60)(58,61)(59,62)(66,78)(67,79)(68,80)(69,75)(70,76)
(71,77)(72,81)(73,82)(74,83);;
s3 := ( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18)(10,20)(11,19)(22,23)
(25,26)(28,29)(30,68)(31,67)(32,66)(33,71)(34,70)(35,69)(36,74)(37,73)(38,72)
(39,59)(40,58)(41,57)(42,62)(43,61)(44,60)(45,65)(46,64)(47,63)(48,77)(49,76)
(50,75)(51,80)(52,79)(53,78)(54,83)(55,82)(56,81);;
s4 := ( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,38)(11,37)(12,48)
(13,50)(14,49)(15,51)(16,53)(17,52)(18,54)(19,56)(20,55)(21,39)(22,41)(23,40)
(24,42)(25,44)(26,43)(27,45)(28,47)(29,46)(57,59)(60,62)(63,65)(66,77)(67,76)
(68,75)(69,80)(70,79)(71,78)(72,83)(73,82)(74,81);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
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, 7)( 5, 8)(12,24)(13,25)(14,26)(15,21)(16,22)(17,23)
(18,27)(19,28)(20,29)(30,33)(31,34)(32,35)(39,51)(40,52)(41,53)(42,48)(43,49)
(44,50)(45,54)(46,55)(47,56)(57,60)(58,61)(59,62)(66,78)(67,79)(68,80)(69,75)
(70,76)(71,77)(72,81)(73,82)(74,83);
s3 := Sym(83)!( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18)(10,20)(11,19)
(22,23)(25,26)(28,29)(30,68)(31,67)(32,66)(33,71)(34,70)(35,69)(36,74)(37,73)
(38,72)(39,59)(40,58)(41,57)(42,62)(43,61)(44,60)(45,65)(46,64)(47,63)(48,77)
(49,76)(50,75)(51,80)(52,79)(53,78)(54,83)(55,82)(56,81);
s4 := Sym(83)!( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,38)(11,37)
(12,48)(13,50)(14,49)(15,51)(16,53)(17,52)(18,54)(19,56)(20,55)(21,39)(22,41)
(23,40)(24,42)(25,44)(26,43)(27,45)(28,47)(29,46)(57,59)(60,62)(63,65)(66,77)
(67,76)(68,75)(69,80)(70,79)(71,78)(72,83)(73,82)(74,81);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope