Polytope of Type {6,12} with group SmallGroup(1296,3528)

Overview

Group: SmallGroup(1296,3528)
Rank: 3
Schläfli Type: {6,12}
Vertices, edges, …: 54, 324, 108
Order of s₀s₁s₂: 12
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

9-fold

18-fold

27-fold

{6,4}*48a

36-fold

{3,6}*36

54-fold

{6,2}*24

81-fold

{2,4}*16

108-fold

{3,2}*12

162-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*s2*(s1*s0)^2*s1*s2> of order 2

63 facets

45 of {6}*12
18 of {3}*6

27 vertex figures

27 of {12}*24

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

54 facets

54 of {6}*12

27 vertex figures

27 of {12}*24

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

54 facets

54 of {6}*12

30 vertex figures

24 of {12}*24
6 of {6}*12

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

36 facets

36 of {6}*12

18 vertex figures

18 of {12}*24

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

36 facets

36 of {6}*12

18 vertex figures

18 of {12}*24

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

36 facets

36 of {6}*12

18 vertex figures

18 of {12}*24

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

36 facets

36 of {6}*12

18 vertex figures

18 of {12}*24

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

36 facets

36 of {6}*12

18 vertex figures

18 of {12}*24

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

36 facets

36 of {6}*12

18 vertex figures

18 of {12}*24

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

36 facets

36 of {6}*12

18 vertex figures

18 of {12}*24

P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1> of order 3

54 facets

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

18 vertex figures

18 of {12}*24

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2> of order 4

36 facets

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

15 vertex figures

12 of {12}*24
3 of {6}*12

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, s0*s1*s2*s1*s0*(s1*s2)^3*s1> of order 6

18 facets

18 of {6}*12

12 vertex figures

6 of {6}*12
6 of {12}*24

P/N, where N=<((s1*s0)^2*s1*s2)^2, s1*s0*s1*s2*s1*s0*(s1*s2)^3> of order 6

18 facets

18 of {6}*12

12 vertex figures

6 of {12}*24
6 of {6}*12

P/N, where N=<(s0*s1)^3*s2*(s1*s0)^2*s1*s2, (s0*s1)^2*s0*s2*(s1*s0)^2*s1*s2*s1> of order 6

18 facets

18 of {6}*12

9 vertex figures

9 of {12}*24

P/N, where N=<s0*s2*(s1*s0)^2*s1*s2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s2> of order 6

21 facets

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

9 vertex figures

9 of {12}*24

P/N, where N=<s0*s2*(s1*s0)^2*s1*s2, s0*s1*s2*(s1*s0)^2*s1*s2*s1> of order 6

21 facets

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

9 vertex figures

9 of {12}*24

P/N, where N=<s0*s2*(s1*s0)^2*s1*s2, s1*s0*s2*(s1*s0)^2*s1*s2*s1> of order 6

27 facets

9 of {6}*12
18 of {3}*6

9 vertex figures

9 of {12}*24

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9

18 facets

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

6 vertex figures

6 of {12}*24

P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 9

12 facets

12 of {6}*12

6 vertex figures

6 of {12}*24

P/N, where N=<(s0*s1)^2, (s2*s1*s0)^2*(s1*s2)^2> of order 9

24 facets

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

6 vertex figures

6 of {12}*24

P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^2> of order 9

24 facets

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

6 vertex figures

6 of {12}*24

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, (s0*s1)^3*(s2*s1)^2*s0*s2*s1*s2> of order 9

12 facets

12 of {6}*12

6 vertex figures

6 of {12}*24

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2> of order 9

12 facets

12 of {6}*12

6 vertex figures

6 of {12}*24

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2, s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1> of order 9

12 facets

12 of {6}*12

6 vertex figures

6 of {12}*24

P/N, where N=<(s0*s1)^2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9

18 facets

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

6 vertex figures

6 of {12}*24

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, s0*s1*s0*s2*s1*s0*(s1*s2)^3> of order 12

15 facets

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

6 vertex figures

3 of {12}*24
3 of {6}*12

P/N, where N=<(s0*s1)^2, s0*s2*(s1*s0)^2*s1*s2> of order 18

15 facets

9 of {2}*4
6 of {3}*6

3 vertex figures

3 of {12}*24

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.

Overview

Special Properties

Related Polytopes

Quotients maximal quotients in bold

Covers minimal covers in bold

Irregular Quotients of which this is a minimal cover

Representations

References

Twisty Puzzle