Polytope of Type {6,3,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,6,3}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2344)
Rank : 5
Schlafli Type : {6,3,6,3}
Number of vertices, edges, etc : 6, 27, 27, 27, 3
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   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) :
   3-fold quotients : {2,3,6,3}*648b, {6,3,6,3}*648
   9-fold quotients : {2,3,6,3}*216, {6,3,2,3}*216
   27-fold quotients : {2,3,2,3}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)
(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)
(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);;
s1 := ( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,37)
(11,38)(12,39)(13,43)(14,44)(15,45)(16,40)(17,41)(18,42)(19,46)(20,47)(21,48)
(22,52)(23,53)(24,54)(25,49)(26,50)(27,51)(58,61)(59,62)(60,63)(67,70)(68,71)
(69,72)(76,79)(77,80)(78,81);;
s2 := ( 4, 8)( 5, 9)( 6, 7)(10,14)(11,15)(12,13)(19,27)(20,25)(21,26)(28,55)
(29,56)(30,57)(31,62)(32,63)(33,61)(34,60)(35,58)(36,59)(37,68)(38,69)(39,67)
(40,66)(41,64)(42,65)(43,70)(44,71)(45,72)(46,81)(47,79)(48,80)(49,76)(50,77)
(51,78)(52,74)(53,75)(54,73);;
s3 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)
(22,25)(23,27)(24,26)(28,37)(29,39)(30,38)(31,43)(32,45)(33,44)(34,40)(35,42)
(36,41)(47,48)(49,52)(50,54)(51,53)(55,64)(56,66)(57,65)(58,70)(59,72)(60,71)
(61,67)(62,69)(63,68)(74,75)(76,79)(77,81)(78,80);;
s4 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)
(16,22)(17,24)(18,23)(29,30)(31,34)(32,36)(33,35)(37,46)(38,48)(39,47)(40,52)
(41,54)(42,53)(43,49)(44,51)(45,50)(56,57)(58,61)(59,63)(60,62)(64,73)(65,75)
(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2, 
s3*s4*s3*s4*s3*s4, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)
(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)
(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);
s1 := Sym(81)!( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)
(10,37)(11,38)(12,39)(13,43)(14,44)(15,45)(16,40)(17,41)(18,42)(19,46)(20,47)
(21,48)(22,52)(23,53)(24,54)(25,49)(26,50)(27,51)(58,61)(59,62)(60,63)(67,70)
(68,71)(69,72)(76,79)(77,80)(78,81);
s2 := Sym(81)!( 4, 8)( 5, 9)( 6, 7)(10,14)(11,15)(12,13)(19,27)(20,25)(21,26)
(28,55)(29,56)(30,57)(31,62)(32,63)(33,61)(34,60)(35,58)(36,59)(37,68)(38,69)
(39,67)(40,66)(41,64)(42,65)(43,70)(44,71)(45,72)(46,81)(47,79)(48,80)(49,76)
(50,77)(51,78)(52,74)(53,75)(54,73);
s3 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)
(20,21)(22,25)(23,27)(24,26)(28,37)(29,39)(30,38)(31,43)(32,45)(33,44)(34,40)
(35,42)(36,41)(47,48)(49,52)(50,54)(51,53)(55,64)(56,66)(57,65)(58,70)(59,72)
(60,71)(61,67)(62,69)(63,68)(74,75)(76,79)(77,81)(78,80);
s4 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)
(15,26)(16,22)(17,24)(18,23)(29,30)(31,34)(32,36)(33,35)(37,46)(38,48)(39,47)
(40,52)(41,54)(42,53)(43,49)(44,51)(45,50)(56,57)(58,61)(59,63)(60,62)(64,73)
(65,75)(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);
poly := sub<Sym(81)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s1*s2*s1*s2*s1*s2, s3*s4*s3*s4*s3*s4, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3 >;

References : None.
to this polytope