Polytope of Type {27,6}

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

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

Permutation Representation (Magma) :

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

References : None.
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