Polytope of Type {10,6,4} with group SmallGroup(1440,5890)

Overview

Group: SmallGroup(1440,5890)
Rank: 4
Schläfli Type: {10,6,4}
Vertices, edges, …: 10, 90, 36, 12
Order of s₀s₁s₂s₃: 20
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{10,6|2},{6,4}₄}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

5-fold

{2,6,4}*288

9-fold

{10,2,4}*160

10-fold

{2,6,4}*144

18-fold

36-fold

{5,2,2}*40

45-fold

{2,2,4}*32

90-fold

{2,2,2}*16

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s2*s3*s2)^2> of order 3

4 facets

4 of {10,6}*120

10 vertex figures

10 of 3-fold non-regular quotient of {6,4}*144

P/N, where N=<(s1*s2)^2> of order 3

8 facets

6 of {10,2}*40
2 of {10,6}*120

10 vertex figures

10 of 3-fold non-regular quotient of {6,4}*144

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

References

None.

to this polytope.