Polytope of Type {10,6,6} with group SmallGroup(720,813)

Overview

Group: SmallGroup(720,813)
Rank: 4
Schläfli Type: {10,6,6}
Vertices, edges, …: 10, 30, 18, 6
Order of s₀s₁s₂s₃: 30
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{10,6|2},{6,6|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

5-fold

{2,6,6}*144a

6-fold

9-fold

{10,2,2}*80

12-fold

{5,2,3}*60

15-fold

18-fold

{5,2,2}*40

30-fold

45-fold

{2,2,2}*16

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := ( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89);;
s1 := ( 1, 2)( 3, 5)( 6,12)( 7,11)( 8,15)( 9,14)(10,13)(16,17)(18,20)(21,27)(22,26)(23,30)(24,29)(25,28)(31,32)(33,35)(36,42)(37,41)(38,45)(39,44)(40,43)(46,47)(48,50)(51,57)(52,56)(53,60)(54,59)(55,58)(61,62)(63,65)(66,72)(67,71)(68,75)(69,74)(70,73)(76,77)(78,80)(81,87)(82,86)(83,90)(84,89)(85,88);;
s2 := ( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(16,36)(17,37)(18,38)(19,39)(20,40)(21,31)(22,32)(23,33)(24,34)(25,35)(26,41)(27,42)(28,43)(29,44)(30,45)(46,51)(47,52)(48,53)(49,54)(50,55)(61,81)(62,82)(63,83)(64,84)(65,85)(66,76)(67,77)(68,78)(69,79)(70,80)(71,86)(72,87)(73,88)(74,89)(75,90);;
s3 := ( 1,61)( 2,62)( 3,63)( 4,64)( 5,65)( 6,66)( 7,67)( 8,68)( 9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(90)!( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89);
s1 := Sym(90)!( 1, 2)( 3, 5)( 6,12)( 7,11)( 8,15)( 9,14)(10,13)(16,17)(18,20)(21,27)(22,26)(23,30)(24,29)(25,28)(31,32)(33,35)(36,42)(37,41)(38,45)(39,44)(40,43)(46,47)(48,50)(51,57)(52,56)(53,60)(54,59)(55,58)(61,62)(63,65)(66,72)(67,71)(68,75)(69,74)(70,73)(76,77)(78,80)(81,87)(82,86)(83,90)(84,89)(85,88);
s2 := Sym(90)!( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(16,36)(17,37)(18,38)(19,39)(20,40)(21,31)(22,32)(23,33)(24,34)(25,35)(26,41)(27,42)(28,43)(29,44)(30,45)(46,51)(47,52)(48,53)(49,54)(50,55)(61,81)(62,82)(63,83)(64,84)(65,85)(66,76)(67,77)(68,78)(69,79)(70,80)(71,86)(72,87)(73,88)(74,89)(75,90);
s3 := Sym(90)!( 1,61)( 2,62)( 3,63)( 4,64)( 5,65)( 6,66)( 7,67)( 8,68)( 9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90);
poly := sub<Sym(90)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

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

References