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

Overview

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

Special Properties

Universal
Orientable
Flat
Self-Dual

Quotients maximal quotients in bold

3-fold

5-fold

{6,2,6}*144

9-fold

{2,10,2}*80

10-fold

15-fold

18-fold

{2,5,2}*40

20-fold

{3,2,3}*36

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

References