Polytope of Type {6,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,27}*324
if this polytope has a name.
Group : SmallGroup(324,38)
Rank : 3
Schlafli Type : {6,27}
Number of vertices, edges, etc : 6, 81, 27
Order of s₀s₁s₂ : 54
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,27,2} of size 648
   {6,27,4} of size 1296
   {6,27,6} of size 1944
Vertex Figure Of :
   {2,6,27} of size 648
   {3,6,27} of size 972
   {4,6,27} of size 1296
   {6,6,27} of size 1944
   {6,6,27} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,27}*108, {6,9}*108
   9-fold quotients : {2,9}*36, {6,3}*36
   27-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,54}*648b
   3-fold covers : {18,27}*972, {6,27}*972a, {6,81}*972
   4-fold covers : {6,108}*1296b, {12,54}*1296b, {6,27}*1296, {12,27}*1296
   5-fold covers : {6,135}*1620
   6-fold covers : {18,54}*1944b, {6,54}*1944a, {6,162}*1944b, {6,54}*1944g
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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