Polytope of Type {6,10,3} with group SmallGroup(1800,575)

Overview

Group: SmallGroup(1800,575)
Rank: 4
Schläfli Type: {6,10,3}
Vertices, edges, …: 6, 150, 75, 15
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{6,10|2},{10,3}₆}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

{2,10,3}*600

25-fold

{6,2,3}*72

50-fold

{3,2,3}*36

75-fold

{2,2,3}*24

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)^2> of order 5

7 facets

5 of {6,2}*24
2 of {6,10}*120

6 vertex figures

6 of 5-fold non-regular quotient of {10,3}*300

P/N, where N=<(s1*s2)^2*s1*s3*s2*s1*s2*s3> of order 5

3 facets

3 of {6,10}*120

6 vertex figures

6 of 5-fold non-regular quotient of {10,3}*300

Representations

Permutation Representation (GAP)

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

References

None.

to this polytope.