Overview
- Group
- SmallGroup(1296,3490)
- Rank
- 3
- Schläfli Type
- {6,4}
- Vertices, edges, …
- 162, 324, 108
- Order of s0s1s2
- 9
- Order of s0s1s2s1
- 4
- Also known as
- if this polytope has a name.
Special Properties
- Compact Hyperbolic Quotient
- Locally Spherical
- Non-Orientable
Quotients maximal quotients in bold
27-fold
54-fold
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*s0*(s2*(s1*s0)^2*s1)^2*s2*s1*s0*s1> of order 2
54 facets
- 54 of {6}*12
84 vertex figures
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 3
36 facets
- 36 of {6}*12
54 vertex figures
- 54 of {4}*8
P/N, where N=<s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 3
36 facets
- 36 of {6}*12
54 vertex figures
- 54 of {4}*8
P/N, where N=<s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s1> of order 3
36 facets
- 36 of {6}*12
54 vertex figures
- 54 of {4}*8
P/N, where N=<(s0*s1)^3*(s2*(s1*s0)^2*s1)^2*s2, (s0*s1)^2*s0*(s2*(s1*s0)^2*s1)^2*s2*s1> of order 4
27 facets
- 27 of {6}*12
45 vertex figures
P/N, where N=<s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s1, s1*s2*(s1*s0)^2*s1*(s2*s1*s0)^2*s1*s2*s1> of order 6
18 facets
- 18 of {6}*12
30 vertex figures
P/N, where N=<(s1*s2)^2, (s0*s1)^3*s2*s1*s0*s2*(s1*s0)^2> of order 6
18 facets
- 18 of {6}*12
30 vertex figures
P/N, where N=<(s0*s1)^3, s0*s1*s0*s2*(s1*s0)^2*s1*s2*s1*s0> of order 6
20 facets
27 vertex figures
- 27 of {4}*8
P/N, where N=<(s0*s1)^3, s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 6
20 facets
27 vertex figures
- 27 of {4}*8
P/N, where N=<(s0*s1)^3, s1*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 6
20 facets
27 vertex figures
- 27 of {4}*8
P/N, where N=<s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2, (s0*s1)^2*s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9
12 facets
- 12 of {6}*12
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s0*s1)^2, s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 9
14 facets
18 vertex figures
- 18 of {4}*8
P/N, where N=<s0*(s1*s0*s2)^2*s1*s2, (s1*s0)^2*s2*s1*s0*(s1*s2)^2> of order 9
12 facets
- 12 of {6}*12
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s1*s2*s1, ((s1*s0)^2*s1*s2)^2> of order 9
12 facets
- 12 of {6}*12
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s0*s1)^2, s0*(s1*s2*(s1*s0)^2)^2*s1*s2> of order 12
11 facets
15 vertex figures
P/N, where N=<(s1*s2)^2, (s0*s1)^2*s2*(s1*s0)^2*s2, ((s1*s0)^2*s1*s2)^2> of order 18
6 facets
- 6 of {6}*12
12 vertex figures
P/N, where N=<(s0*s1)^3, s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2, s1*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 18
8 facets
9 vertex figures
- 9 of {4}*8
P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, s1*s0*s2*(s1*s0)^2*s1*s2*s1> of order 18
8 facets
9 vertex figures
- 9 of {4}*8
P/N, where N=<(s0*s1)^3, s0*(s1*s0*s2)^2*s1*s2, s0*s1*s2*s1*s0*(s1*s2)^2> of order 18
8 facets
9 vertex figures
- 9 of {4}*8
Representations
Permutation Representation (GAP)
s0 := ( 4,10)( 5,12)( 6,11);; s1 := ( 2, 3)( 5, 6)( 7,10)( 8,12)( 9,11);; s2 := ( 1, 9)( 2, 7)( 3, 8)( 4,11)( 5,12)( 6,10);; 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,
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 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(12)!( 4,10)( 5,12)( 6,11); s1 := Sym(12)!( 2, 3)( 5, 6)( 7,10)( 8,12)( 9,11); s2 := Sym(12)!( 1, 9)( 2, 7)( 3, 8)( 4,11)( 5,12)( 6,10); poly := sub<Sym(12)|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, 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 >;
References
None.
to this polytope.