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

Overview

Group: SmallGroup(1800,575)
Rank: 4
Schläfli Type: {3,6,6}
Vertices, edges, …: 25, 75, 150, 6
Order of s₀s₁s₂s₃: 30
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{3,6}₁₀,{6,6|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

{3,6,2}*600

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=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 5

6 facets

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

5 vertex figures

5 of {6,6}*72a

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.