Polytope of Type {2,3,6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,18}*1296b
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 5
Schlafli Type : {2,3,6,18}
Number of vertices, edges, etc : 2, 3, 9, 54, 18
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 : {2,3,2,18}*432, {2,3,6,6}*432b
6-fold quotients : {2,3,2,9}*216
9-fold quotients : {2,3,2,6}*144, {2,3,6,2}*144
18-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 6, 9)( 7,10)( 8,11)(12,21)(13,22)(14,23)(15,27)(16,28)(17,29)(18,24)
(19,25)(20,26)(33,36)(34,37)(35,38)(39,48)(40,49)(41,50)(42,54)(43,55)(44,56)
(45,51)(46,52)(47,53)(60,63)(61,64)(62,65)(66,75)(67,76)(68,77)(69,81)(70,82)
(71,83)(72,78)(73,79)(74,80);;
s2 := ( 3,15)( 4,16)( 5,17)( 6,12)( 7,13)( 8,14)( 9,18)(10,19)(11,20)(21,24)
(22,25)(23,26)(30,42)(31,43)(32,44)(33,39)(34,40)(35,41)(36,45)(37,46)(38,47)
(48,51)(49,52)(50,53)(57,69)(58,70)(59,71)(60,66)(61,67)(62,68)(63,72)(64,73)
(65,74)(75,78)(76,79)(77,80);;
s3 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)(24,27)
(25,29)(26,28)(30,59)(31,58)(32,57)(33,65)(34,64)(35,63)(36,62)(37,61)(38,60)
(39,68)(40,67)(41,66)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,77)(49,76)
(50,75)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78);;
s4 := ( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,38)(11,37)(12,39)
(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,47)(20,46)(21,48)(22,50)(23,49)
(24,51)(25,53)(26,52)(27,54)(28,56)(29,55)(57,59)(60,62)(63,65)(66,68)(69,71)
(72,74)(75,77)(78,80)(81,83);;
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,
s1*s2*s1*s2*s1*s2, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*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)(12,21)(13,22)(14,23)(15,27)(16,28)(17,29)
(18,24)(19,25)(20,26)(33,36)(34,37)(35,38)(39,48)(40,49)(41,50)(42,54)(43,55)
(44,56)(45,51)(46,52)(47,53)(60,63)(61,64)(62,65)(66,75)(67,76)(68,77)(69,81)
(70,82)(71,83)(72,78)(73,79)(74,80);
s2 := Sym(83)!( 3,15)( 4,16)( 5,17)( 6,12)( 7,13)( 8,14)( 9,18)(10,19)(11,20)
(21,24)(22,25)(23,26)(30,42)(31,43)(32,44)(33,39)(34,40)(35,41)(36,45)(37,46)
(38,47)(48,51)(49,52)(50,53)(57,69)(58,70)(59,71)(60,66)(61,67)(62,68)(63,72)
(64,73)(65,74)(75,78)(76,79)(77,80);
s3 := Sym(83)!( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)
(24,27)(25,29)(26,28)(30,59)(31,58)(32,57)(33,65)(34,64)(35,63)(36,62)(37,61)
(38,60)(39,68)(40,67)(41,66)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,77)
(49,76)(50,75)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78);
s4 := Sym(83)!( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,38)(11,37)
(12,39)(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,47)(20,46)(21,48)(22,50)
(23,49)(24,51)(25,53)(26,52)(27,54)(28,56)(29,55)(57,59)(60,62)(63,65)(66,68)
(69,71)(72,74)(75,77)(78,80)(81,83);
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, s1*s2*s1*s2*s1*s2,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope