Polytope of Type {3,2,3,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,3,6,8}*1728
if this polytope has a name.
Group : SmallGroup(1728,37593)
Rank : 6
Schlafli Type : {3,2,3,6,8}
Number of vertices, edges, etc : 3, 3, 3, 9, 24, 8
Order of s0s1s2s3s4s5 : 24
Order of s0s1s2s3s4s5s4s3s2s1 : 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) :
2-fold quotients : {3,2,3,6,4}*864
3-fold quotients : {3,2,3,2,8}*576
4-fold quotients : {3,2,3,6,2}*432
6-fold quotients : {3,2,3,2,4}*288
12-fold quotients : {3,2,3,2,2}*144
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(23,24)(25,28)
(26,30)(27,29)(32,33)(34,37)(35,39)(36,38)(41,42)(43,46)(44,48)(45,47)(50,51)
(52,55)(53,57)(54,56)(59,60)(61,64)(62,66)(63,65)(68,69)(70,73)(71,75)
(72,74);;
s3 := ( 4, 8)( 5, 7)( 6, 9)(10,11)(13,17)(14,16)(15,18)(19,20)(22,26)(23,25)
(24,27)(28,29)(31,35)(32,34)(33,36)(37,38)(40,44)(41,43)(42,45)(46,47)(49,53)
(50,52)(51,54)(55,56)(58,62)(59,61)(60,63)(64,65)(67,71)(68,70)(69,72)
(73,74);;
s4 := ( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(22,31)(23,32)(24,33)(25,37)
(26,38)(27,39)(28,34)(29,35)(30,36)(40,58)(41,59)(42,60)(43,64)(44,65)(45,66)
(46,61)(47,62)(48,63)(49,67)(50,68)(51,69)(52,73)(53,74)(54,75)(55,70)(56,71)
(57,72);;
s5 := ( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)(11,47)(12,48)(13,49)
(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,67)(23,68)(24,69)
(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,58)(32,59)(33,60)(34,61)(35,62)
(36,63)(37,64)(38,65)(39,66);;
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, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s0*s5*s0*s5, s1*s5*s1*s5,
s2*s5*s2*s5, s3*s5*s3*s5, s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3, s3*s4*s5*s4*s3*s4*s5*s4,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(75)!(2,3);
s1 := Sym(75)!(1,2);
s2 := Sym(75)!( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(23,24)
(25,28)(26,30)(27,29)(32,33)(34,37)(35,39)(36,38)(41,42)(43,46)(44,48)(45,47)
(50,51)(52,55)(53,57)(54,56)(59,60)(61,64)(62,66)(63,65)(68,69)(70,73)(71,75)
(72,74);
s3 := Sym(75)!( 4, 8)( 5, 7)( 6, 9)(10,11)(13,17)(14,16)(15,18)(19,20)(22,26)
(23,25)(24,27)(28,29)(31,35)(32,34)(33,36)(37,38)(40,44)(41,43)(42,45)(46,47)
(49,53)(50,52)(51,54)(55,56)(58,62)(59,61)(60,63)(64,65)(67,71)(68,70)(69,72)
(73,74);
s4 := Sym(75)!( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(22,31)(23,32)(24,33)
(25,37)(26,38)(27,39)(28,34)(29,35)(30,36)(40,58)(41,59)(42,60)(43,64)(44,65)
(45,66)(46,61)(47,62)(48,63)(49,67)(50,68)(51,69)(52,73)(53,74)(54,75)(55,70)
(56,71)(57,72);
s5 := Sym(75)!( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)(11,47)(12,48)
(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,67)(23,68)
(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,58)(32,59)(33,60)(34,61)
(35,62)(36,63)(37,64)(38,65)(39,66);
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, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s5*s0*s5,
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5,
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3,
s3*s4*s5*s4*s3*s4*s5*s4, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3,
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;
to this polytope