Polytope of Type {6,30} with group SmallGroup(1800,575)

Overview

Group: SmallGroup(1800,575)
Rank: 3
Schläfli Type: {6,30}
Vertices, edges, …: 30, 450, 150
Order of s₀s₁s₂: 6
Order of s₀s₁s₂s₁: 10
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

{6,10}*600b

6-fold

{3,10}*300

25-fold

{6,6}*72a

75-fold

150-fold

225-fold

{2,2}*8

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)^2> of order 5

30 facets

30 of {6}*12

14 vertex figures

4 of {30}*60
10 of {6}*12

P/N, where N=<s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 5

30 facets

30 of {6}*12

6 vertex figures

6 of {30}*60

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

Finitely Presented Group Representation (GAP)

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

Finitely Presented Group Representation (Magma)

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

References

None.

to this polytope.