Polytope of Type {3,6,4,2,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,4,2,3}*864
if this polytope has a name.
Group : SmallGroup(864,4406)
Rank : 6
Schlafli Type : {3,6,4,2,3}
Number of vertices, edges, etc : 3, 9, 12, 4, 3, 3
Order of s0s1s2s3s4s5 : 12
Order of s0s1s2s3s4s5s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,6,4,2,3,2} of size 1728
Vertex Figure Of :
{2,3,6,4,2,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6,2,2,3}*432
3-fold quotients : {3,2,4,2,3}*288
6-fold quotients : {3,2,2,2,3}*144
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,6,8,2,3}*1728, {3,6,4,2,6}*1728, {6,6,4,2,3}*1728c
Permutation Representation (GAP) :
s0 := ( 1,37)( 2,39)( 3,38)( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)(10,46)
(11,48)(12,47)(13,52)(14,54)(15,53)(16,49)(17,51)(18,50)(19,55)(20,57)(21,56)
(22,61)(23,63)(24,62)(25,58)(26,60)(27,59)(28,64)(29,66)(30,65)(31,70)(32,72)
(33,71)(34,67)(35,69)(36,68);;
s1 := ( 1,41)( 2,40)( 3,42)( 4,38)( 5,37)( 6,39)( 7,44)( 8,43)( 9,45)(10,50)
(11,49)(12,51)(13,47)(14,46)(15,48)(16,53)(17,52)(18,54)(19,59)(20,58)(21,60)
(22,56)(23,55)(24,57)(25,62)(26,61)(27,63)(28,68)(29,67)(30,69)(31,65)(32,64)
(33,66)(34,71)(35,70)(36,72);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(19,28)(20,30)(21,29)(22,31)
(23,33)(24,32)(25,34)(26,36)(27,35)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)
(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71);;
s3 := ( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)
(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)
(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)
(51,69)(52,70)(53,71)(54,72);;
s4 := (74,75);;
s5 := (73,74);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;; s5 := F.6;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s3*s4*s3*s4, s0*s5*s0*s5, s1*s5*s1*s5,
s2*s5*s2*s5, s3*s5*s3*s5, s0*s1*s0*s1*s0*s1,
s4*s5*s4*s5*s4*s5, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(75)!( 1,37)( 2,39)( 3,38)( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)
(10,46)(11,48)(12,47)(13,52)(14,54)(15,53)(16,49)(17,51)(18,50)(19,55)(20,57)
(21,56)(22,61)(23,63)(24,62)(25,58)(26,60)(27,59)(28,64)(29,66)(30,65)(31,70)
(32,72)(33,71)(34,67)(35,69)(36,68);
s1 := Sym(75)!( 1,41)( 2,40)( 3,42)( 4,38)( 5,37)( 6,39)( 7,44)( 8,43)( 9,45)
(10,50)(11,49)(12,51)(13,47)(14,46)(15,48)(16,53)(17,52)(18,54)(19,59)(20,58)
(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)(27,63)(28,68)(29,67)(30,69)(31,65)
(32,64)(33,66)(34,71)(35,70)(36,72);
s2 := Sym(75)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(19,28)(20,30)(21,29)
(22,31)(23,33)(24,32)(25,34)(26,36)(27,35)(38,39)(41,42)(44,45)(47,48)(50,51)
(53,54)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71);
s3 := Sym(75)!( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)
(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)
(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)
(50,68)(51,69)(52,70)(53,71)(54,72);
s4 := Sym(75)!(74,75);
s5 := Sym(75)!(73,74);
poly := sub<Sym(75)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s5*s5, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4, s0*s5*s0*s5,
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5,
s0*s1*s0*s1*s0*s1, s4*s5*s4*s5*s4*s5,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;
to this polytope