Polytope of Type {6,3,6,9} with group SmallGroup(1944,2340)

Overview

Group: SmallGroup(1944,2340)
Rank: 5
Schläfli Type: {6,3,6,9}
Vertices, edges, …: 6, 9, 9, 27, 9
Order of s₀s₁s₂s₃s₄: 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

{2,3,2,3}*72

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References