Polytope of Type {21,6}

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

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

Permutation Representation (Magma) :

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

References : None.
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