Polytope of Type {3,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,9}*324
if this polytope has a name.
Group : SmallGroup(324,37)
Rank : 4
Schlafli Type : {3,6,9}
Number of vertices, edges, etc : 3, 9, 27, 9
Order of s₀s₁s₂s₃ : 9
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,9,2} of size 648
   {3,6,9,4} of size 1296
   {3,6,9,6} of size 1944
Vertex Figure Of :
   {2,3,6,9} of size 648
   {4,3,6,9} of size 1296
   {6,3,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,9}*108, {3,6,3}*108
   9-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,18}*648a, {6,6,9}*648a
   3-fold covers : {9,6,9}*972, {3,6,27}*972, {3,6,9}*972a, {3,6,9}*972b
   4-fold covers : {12,6,9}*1296a, {3,6,36}*1296a, {6,6,18}*1296a
   5-fold covers : {3,6,45}*1620, {15,6,9}*1620
   6-fold covers : {9,6,18}*1944a, {18,6,9}*1944a, {3,6,54}*1944a, {6,6,27}*1944a, {3,6,18}*1944a, {6,6,9}*1944a, {3,6,18}*1944b, {6,6,9}*1944b, {3,6,18}*1944d, {6,6,9}*1944d
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)
(17,27)(18,26)(29,30)(32,33)(35,36)(37,46)(38,48)(39,47)(40,49)(41,51)(42,50)
(43,52)(44,54)(45,53)(56,57)(59,60)(62,63)(64,73)(65,75)(66,74)(67,76)(68,78)
(69,77)(70,79)(71,81)(72,80);;
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)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)(36,44)
(47,48)(50,51)(53,54)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)
(63,71)(74,75)(77,78)(80,81);;
s2 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,38)
(11,37)(12,39)(13,44)(14,43)(15,45)(16,41)(17,40)(18,42)(19,48)(20,47)(21,46)
(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(55,58)(56,60)(57,59)(62,63)(64,68)
(65,67)(66,69)(70,71)(73,78)(74,77)(75,76)(79,81);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)
(23,27)(24,26)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,61)(35,63)(36,62)
(37,67)(38,69)(39,68)(40,64)(41,66)(42,65)(43,70)(44,72)(45,71)(46,76)(47,78)
(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
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(81)!( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)
(16,25)(17,27)(18,26)(29,30)(32,33)(35,36)(37,46)(38,48)(39,47)(40,49)(41,51)
(42,50)(43,52)(44,54)(45,53)(56,57)(59,60)(62,63)(64,73)(65,75)(66,74)(67,76)
(68,78)(69,77)(70,79)(71,81)(72,80);
s1 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)
(20,21)(23,24)(26,27)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)
(36,44)(47,48)(50,51)(53,54)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)
(62,72)(63,71)(74,75)(77,78)(80,81);
s2 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,38)(11,37)(12,39)(13,44)(14,43)(15,45)(16,41)(17,40)(18,42)(19,48)(20,47)
(21,46)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(55,58)(56,60)(57,59)(62,63)
(64,68)(65,67)(66,69)(70,71)(73,78)(74,77)(75,76)(79,81);
s3 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)
(22,25)(23,27)(24,26)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,61)(35,63)
(36,62)(37,67)(38,69)(39,68)(40,64)(41,66)(42,65)(43,70)(44,72)(45,71)(46,76)
(47,78)(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);
poly := sub<Sym(81)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope