Polytope of Type {6,21}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,21}*252
if this polytope has a name.
Group : SmallGroup(252,36)
Rank : 3
Schlafli Type : {6,21}
Number of vertices, edges, etc : 6, 63, 21
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,21,2} of size 504
   {6,21,4} of size 1008
   {6,21,6} of size 1512
Vertex Figure Of :
   {2,6,21} of size 504
   {3,6,21} of size 756
   {4,6,21} of size 1008
   {6,6,21} of size 1512
   {6,6,21} of size 1512
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,21}*84
   7-fold quotients : {6,3}*36
   9-fold quotients : {2,7}*28
   21-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,42}*504c
   3-fold covers : {6,63}*756, {6,21}*756
   4-fold covers : {6,84}*1008c, {12,42}*1008c, {12,21}*1008, {6,21}*1008b
   5-fold covers : {6,105}*1260
   6-fold covers : {6,126}*1512b, {6,42}*1512b, {6,42}*1512d
   7-fold covers : {6,147}*1764, {42,21}*1764
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(63)!(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)
(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)
(42,63);
s1 := Sym(63)!( 1,22)( 2,28)( 3,27)( 4,26)( 5,25)( 6,24)( 7,23)( 8,36)( 9,42)
(10,41)(11,40)(12,39)(13,38)(14,37)(15,29)(16,35)(17,34)(18,33)(19,32)(20,31)
(21,30)(44,49)(45,48)(46,47)(50,57)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58);
s2 := Sym(63)!( 1, 9)( 2, 8)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)(15,16)(17,21)
(18,20)(22,51)(23,50)(24,56)(25,55)(26,54)(27,53)(28,52)(29,44)(30,43)(31,49)
(32,48)(33,47)(34,46)(35,45)(36,58)(37,57)(38,63)(39,62)(40,61)(41,60)(42,59);
poly := sub<Sym(63)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

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

References : None.
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