Polytope of Type {6,4} with group SmallGroup(384,17948)

Overview

Group: SmallGroup(384,17948)
Rank: 3
Schläfli Type: {6,4}
Vertices, edges, …: 48, 96, 32
Order of s₀s₁s₂: 6
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie

Quotients maximal quotients in bold

2-fold

{6,4}*192a

4-fold

{6,4}*96

8-fold

16-fold

32-fold

{3,2}*12

48-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

5-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

16 facets

16 of {6}*12

24 vertex figures

24 of {4}*8

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

16 facets

16 of {6}*12

28 vertex figures

20 of {4}*8
8 of {2}*4

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

16 facets

16 of {6}*12

24 vertex figures

24 of {4}*8

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

20 facets

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

24 vertex figures

24 of {4}*8

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

16 facets

16 of {6}*12

24 vertex figures

24 of {4}*8

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

16 facets

16 of {6}*12

24 vertex figures

24 of {4}*8

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

16 facets

16 of {6}*12

24 vertex figures

24 of {4}*8

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

16 facets

16 of {6}*12

24 vertex figures

24 of {4}*8

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

8 facets

8 of {6}*12

14 vertex figures

10 of {4}*8
4 of {2}*4

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

10 facets

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

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

14 vertex figures

10 of {4}*8
4 of {2}*4

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 4

8 facets

8 of {6}*12

18 vertex figures

6 of {4}*8
12 of {2}*4

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

8 facets

8 of {6}*12

16 vertex figures

8 of {4}*8
8 of {2}*4

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

12 facets

8 of {3}*6
4 of {6}*12

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

14 vertex figures

10 of {4}*8
4 of {2}*4

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

8 facets

8 of {6}*12

14 vertex figures

10 of {4}*8
4 of {2}*4

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

8 facets

8 of {6}*12

12 vertex figures

12 of {4}*8

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

8 facets

8 of {6}*12

16 vertex figures

8 of {4}*8
8 of {2}*4

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

8 facets

8 of {6}*12

14 vertex figures

10 of {4}*8
4 of {2}*4

P/N, where N=<(s0*s1)^3, s0*(s1*s0*s2)^2*s1> of order 8

6 facets

4 of {3}*6
2 of {6}*12

8 vertex figures

4 of {4}*8
4 of {2}*4

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*s2*s1*s0*s2*s1*s2> of order 8

4 facets

4 of {6}*12

6 vertex figures

6 of {4}*8

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1*s0)^2, (s0*s1)^2*s0*s2*s1*s0*(s1*s2)^2> of order 8

4 facets

4 of {6}*12

8 vertex figures

4 of {4}*8
4 of {2}*4

P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*s1*s2)^2, (s0*s1)^2*s2*(s1*s0)^2*s2> of order 8

4 facets

4 of {6}*12

8 vertex figures

4 of {4}*8
4 of {2}*4

P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*s1*s2*s1*s0*s2*s1*s2, (s0*s1)^2*(s2*s1*s0)^2*s2> of order 8

4 facets

4 of {6}*12

6 vertex figures

6 of {4}*8

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, s0*s1*s0*s2*(s1*s0)^2*(s1*s2)^2> of order 8

6 facets

4 of {3}*6
2 of {6}*12

6 vertex figures

6 of {4}*8

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1*s0)^2, (s0*s1)^2*s0*s2*s1*s0*s1*s2*s1> of order 8

4 facets

4 of {6}*12

8 vertex figures

4 of {4}*8
4 of {2}*4

P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1, (s0*s1)^2*s2*(s1*s0)^2*s2> of order 8

4 facets

4 of {6}*12

10 vertex figures

2 of {4}*8
8 of {2}*4

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2, (s0*s1)^2*(s2*s1*s0)^2*s2, s0*s1*s0*s2*(s1*s0)^2*s1*s2*s1> of order 8

4 facets

4 of {6}*12

6 vertex figures

6 of {4}*8

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, s0*(s1*s0*s2)^2*s1*s2> of order 8

6 facets

4 of {3}*6
2 of {6}*12

6 vertex figures

6 of {4}*8

P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*s1*s2*s1*s0*s2*s1*s2, (s0*s1)^2*s2*(s1*s0)^2*s2> of order 8

4 facets

4 of {6}*12

7 vertex figures

5 of {4}*8
2 of {2}*4

P/N, where N=<(s0*s1)^2*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*(s1*s0)^2*s1*s2*s1> of order 8

4 facets

4 of {6}*12

6 vertex figures

6 of {4}*8

Representations

Permutation Representation (GAP)

s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,33)(18,34)(19,36)(20,35)(21,41)(22,42)(23,44)(24,43)(25,37)(26,38)(27,40)(28,39)(29,45)(30,46)(31,48)(32,47)(51,52)(53,57)(54,58)(55,60)(56,59)(63,64)(65,81)(66,82)(67,84)(68,83)(69,89)(70,90)(71,92)(72,91)(73,85)(74,86)(75,88)(76,87)(77,93)(78,94)(79,96)(80,95);;
s1 := ( 1,17)( 2,20)( 3,19)( 4,18)( 5,23)( 6,22)( 7,21)( 8,24)( 9,30)(10,31)(11,32)(12,29)(13,28)(14,25)(15,26)(16,27)(34,36)(37,39)(41,46)(42,47)(43,48)(44,45)(49,65)(50,68)(51,67)(52,66)(53,71)(54,70)(55,69)(56,72)(57,78)(58,79)(59,80)(60,77)(61,76)(62,73)(63,74)(64,75)(82,84)(85,87)(89,94)(90,95)(91,96)(92,93);;
s2 := ( 1,61)( 2,62)( 3,63)( 4,64)( 5,57)( 6,58)( 7,59)( 8,60)( 9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,77)(18,78)(19,79)(20,80)(21,73)(22,74)(23,75)(24,76)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,93)(34,94)(35,95)(36,96)(37,89)(38,90)(39,91)(40,92)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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(96)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,33)(18,34)(19,36)(20,35)(21,41)(22,42)(23,44)(24,43)(25,37)(26,38)(27,40)(28,39)(29,45)(30,46)(31,48)(32,47)(51,52)(53,57)(54,58)(55,60)(56,59)(63,64)(65,81)(66,82)(67,84)(68,83)(69,89)(70,90)(71,92)(72,91)(73,85)(74,86)(75,88)(76,87)(77,93)(78,94)(79,96)(80,95);
s1 := Sym(96)!( 1,17)( 2,20)( 3,19)( 4,18)( 5,23)( 6,22)( 7,21)( 8,24)( 9,30)(10,31)(11,32)(12,29)(13,28)(14,25)(15,26)(16,27)(34,36)(37,39)(41,46)(42,47)(43,48)(44,45)(49,65)(50,68)(51,67)(52,66)(53,71)(54,70)(55,69)(56,72)(57,78)(58,79)(59,80)(60,77)(61,76)(62,73)(63,74)(64,75)(82,84)(85,87)(89,94)(90,95)(91,96)(92,93);
s2 := Sym(96)!( 1,61)( 2,62)( 3,63)( 4,64)( 5,57)( 6,58)( 7,59)( 8,60)( 9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,77)(18,78)(19,79)(20,80)(21,73)(22,74)(23,75)(24,76)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,93)(34,94)(35,95)(36,96)(37,89)(38,90)(39,91)(40,92)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84);
poly := sub<Sym(96)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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