Polytope of Type {6,18,2,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18,2,2}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1862)
Rank : 5
Schlafli Type : {6,18,2,2}
Number of vertices, edges, etc : 9, 81, 27, 2, 2
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {6,6,2,2}*432
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 8)( 5, 9)( 6, 7)(10,19)(11,20)(12,21)(13,26)(14,27)(15,25)(16,24)
(17,22)(18,23)(31,35)(32,36)(33,34)(37,46)(38,47)(39,48)(40,53)(41,54)(42,52)
(43,51)(44,49)(45,50)(58,62)(59,63)(60,61)(64,73)(65,74)(66,75)(67,80)(68,81)
(69,79)(70,78)(71,76)(72,77);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)
(23,24)(26,27)(28,66)(29,65)(30,64)(31,69)(32,68)(33,67)(34,72)(35,71)(36,70)
(37,57)(38,56)(39,55)(40,60)(41,59)(42,58)(43,63)(44,62)(45,61)(46,75)(47,74)
(48,73)(49,78)(50,77)(51,76)(52,81)(53,80)(54,79);;
s2 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,40)
(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,53)(20,52)(21,54)
(22,50)(23,49)(24,51)(25,47)(26,46)(27,48)(55,57)(58,63)(59,62)(60,61)(64,69)
(65,68)(66,67)(70,72)(73,79)(74,81)(75,80)(77,78);;
s3 := (82,83);;
s4 := (84,85);;
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, s2*s3*s2*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s3*s4*s3*s4, s0*s1*s2*s0*s1*s2*s0*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(85)!( 4, 8)( 5, 9)( 6, 7)(10,19)(11,20)(12,21)(13,26)(14,27)(15,25)
(16,24)(17,22)(18,23)(31,35)(32,36)(33,34)(37,46)(38,47)(39,48)(40,53)(41,54)
(42,52)(43,51)(44,49)(45,50)(58,62)(59,63)(60,61)(64,73)(65,74)(66,75)(67,80)
(68,81)(69,79)(70,78)(71,76)(72,77);
s1 := Sym(85)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)
(20,21)(23,24)(26,27)(28,66)(29,65)(30,64)(31,69)(32,68)(33,67)(34,72)(35,71)
(36,70)(37,57)(38,56)(39,55)(40,60)(41,59)(42,58)(43,63)(44,62)(45,61)(46,75)
(47,74)(48,73)(49,78)(50,77)(51,76)(52,81)(53,80)(54,79);
s2 := Sym(85)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,40)(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,53)(20,52)
(21,54)(22,50)(23,49)(24,51)(25,47)(26,46)(27,48)(55,57)(58,63)(59,62)(60,61)
(64,69)(65,68)(66,67)(70,72)(73,79)(74,81)(75,80)(77,78);
s3 := Sym(85)!(82,83);
s4 := Sym(85)!(84,85);
poly := sub<Sym(85)|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, s2*s3*s2*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4,
s0*s1*s2*s0*s1*s2*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
to this polytope