Polytope of Type {6,3,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,6,9}*1944
if this polytope has a name.
Group : SmallGroup(1944,2340)
Rank : 5
Schlafli Type : {6,3,6,9}
Number of vertices, edges, etc : 6, 9, 9, 27, 9
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
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 : {2,3,6,9}*648, {6,3,2,9}*648, {6,3,6,3}*648
9-fold quotients : {2,3,2,9}*216, {2,3,6,3}*216, {6,3,2,3}*216
27-fold quotients : {2,3,2,3}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)
(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)
(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);;
s1 := ( 1,28)( 2,30)( 3,29)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)(10,37)
(11,39)(12,38)(13,40)(14,42)(15,41)(16,43)(17,45)(18,44)(19,46)(20,48)(21,47)
(22,49)(23,51)(24,50)(25,52)(26,54)(27,53)(56,57)(59,60)(62,63)(65,66)(68,69)
(71,72)(74,75)(77,78)(80,81);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(10,11)(13,14)(16,17)(19,21)(22,24)(25,27)(28,55)
(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,65)(38,64)(39,66)
(40,68)(41,67)(42,69)(43,71)(44,70)(45,72)(46,75)(47,74)(48,73)(49,78)(50,77)
(51,76)(52,81)(53,80)(54,79);;
s3 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(19,22)
(20,24)(21,23)(26,27)(28,37)(29,39)(30,38)(31,43)(32,45)(33,44)(34,40)(35,42)
(36,41)(46,49)(47,51)(48,50)(53,54)(55,64)(56,66)(57,65)(58,70)(59,72)(60,71)
(61,67)(62,69)(63,68)(73,76)(74,78)(75,77)(80,81);;
s4 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)
(16,25)(17,27)(18,26)(29,30)(31,34)(32,36)(33,35)(37,49)(38,51)(39,50)(40,46)
(41,48)(42,47)(43,52)(44,54)(45,53)(56,57)(58,61)(59,63)(60,62)(64,76)(65,78)
(66,77)(67,73)(68,75)(69,74)(70,79)(71,81)(72,80);;
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, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*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(81)!(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)
(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)
(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);
s1 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)
(10,37)(11,39)(12,38)(13,40)(14,42)(15,41)(16,43)(17,45)(18,44)(19,46)(20,48)
(21,47)(22,49)(23,51)(24,50)(25,52)(26,54)(27,53)(56,57)(59,60)(62,63)(65,66)
(68,69)(71,72)(74,75)(77,78)(80,81);
s2 := Sym(81)!( 2, 3)( 5, 6)( 8, 9)(10,11)(13,14)(16,17)(19,21)(22,24)(25,27)
(28,55)(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,65)(38,64)
(39,66)(40,68)(41,67)(42,69)(43,71)(44,70)(45,72)(46,75)(47,74)(48,73)(49,78)
(50,77)(51,76)(52,81)(53,80)(54,79);
s3 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)
(19,22)(20,24)(21,23)(26,27)(28,37)(29,39)(30,38)(31,43)(32,45)(33,44)(34,40)
(35,42)(36,41)(46,49)(47,51)(48,50)(53,54)(55,64)(56,66)(57,65)(58,70)(59,72)
(60,71)(61,67)(62,69)(63,68)(73,76)(74,78)(75,77)(80,81);
s4 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,22)(11,24)(12,23)(13,19)(14,21)
(15,20)(16,25)(17,27)(18,26)(29,30)(31,34)(32,36)(33,35)(37,49)(38,51)(39,50)
(40,46)(41,48)(42,47)(43,52)(44,54)(45,53)(56,57)(58,61)(59,63)(60,62)(64,76)
(65,78)(66,77)(67,73)(68,75)(69,74)(70,79)(71,81)(72,80);
poly := sub<Sym(81)|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, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s1*s2*s1*s2*s1*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
References : None.
to this polytope