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

Overview

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

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

{6,6}*600b

6-fold

{3,6}*300

25-fold

{6,6}*72b

50-fold

{6,3}*36

75-fold

{2,6}*24

150-fold

{2,3}*12

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

75 facets

75 of {6}*12

78 vertex figures

72 of {6}*12
6 of {3}*6

P/N, where N=<(s0*s1)^3> of order 2

100 facets

50 of {3}*6
50 of {6}*12

75 vertex figures

75 of {6}*12

P/N, where N=<(s0*s1)^2> of order 3

54 facets

6 of {2}*4
48 of {6}*12

50 vertex figures

50 of {6}*12

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

30 facets

30 of {6}*12

30 vertex figures

30 of {6}*12

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

30 facets

30 of {6}*12

30 vertex figures

30 of {6}*12

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s0*s1)^3*s2*s1*s0*s2*s1*s2> of order 10

15 facets

15 of {6}*12

18 vertex figures

12 of {6}*12
6 of {3}*6

P/N, where N=<(s0*s1)^3, s0*(s1*s2)^2*(s1*s0)^2*s2*s1*s2> of order 10

20 facets

10 of {3}*6
10 of {6}*12

15 vertex figures

15 of {6}*12

P/N, where N=<(s1*s2)^3, s0*s1*s0*(s2*s1)^2*(s0*s1)^2*s2> of order 10

15 facets

15 of {6}*12

18 vertex figures

6 of {3}*6
12 of {6}*12

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

References

None.

to this polytope.