Overview
- Group
- SmallGroup(1944,2324)
- Rank
- 3
- Schläfli Type
- {6,12}
- Vertices, edges, …
- 81, 486, 162
- Order of s0s1s2
- 12
- Order of s0s1s2s1
- 6
- Also known as
- if this polytope has a name.
Special Properties
- Compact Hyperbolic Quotient
- Locally Spherical
- Non-Orientable
Quotients maximal quotients in bold
9-fold
27-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*s2*s1> of order 3
54 facets
- 54 of {6}*12
27 vertex figures
- 27 of {12}*24
P/N, where N=<s1*s0*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 3
54 facets
- 54 of {6}*12
27 vertex figures
- 27 of {12}*24
P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 3
54 facets
- 54 of {6}*12
27 vertex figures
- 27 of {12}*24
P/N, where N=<s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 3
54 facets
- 54 of {6}*12
27 vertex figures
- 27 of {12}*24
P/N, where N=<(s0*(s1*s2)^3*s1)^2> of order 3
54 facets
- 54 of {6}*12
27 vertex figures
- 27 of {12}*24
P/N, where N=<s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 3
72 facets
27 vertex figures
- 27 of {12}*24
P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1> of order 9
36 facets
9 vertex figures
- 9 of {12}*24
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9
18 facets
- 18 of {6}*12
9 vertex figures
- 9 of {12}*24
P/N, where N=<(s0*(s1*s2)^2*s1)^2, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9
18 facets
- 18 of {6}*12
9 vertex figures
- 9 of {12}*24
P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s1*s2*s1*s0*s2*s1, s1*s0*s1*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9
18 facets
- 18 of {6}*12
9 vertex figures
- 9 of {12}*24
P/N, where N=<(s0*(s1*s2)^2*s1)^2, (s1*s0*(s1*s2)^2)^2> of order 9
18 facets
- 18 of {6}*12
9 vertex figures
- 9 of {12}*24
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*(s1*s0)^2*s2*s1> of order 9
24 facets
9 vertex figures
- 9 of {12}*24
P/N, where N=<s1*s0*(s1*s2)^3*s1*s0*s2*s1, (s0*s1)^3*s2*(s1*s0)^2*s1*s2> of order 9
18 facets
- 18 of {6}*12
15 vertex figures
P/N, where N=<(s0*s1)^2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9
30 facets
9 vertex figures
- 9 of {12}*24
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);; s1 := ( 2, 3)( 4, 5)( 7, 9)(10,26)(11,25)(12,27)(13,21)(14,20)(15,19)(16,22)(17,24)(18,23);; s2 := ( 1,10)( 2,17)( 3,15)( 4,16)( 5,14)( 6,12)( 7,13)( 8,11)( 9,18)(20,26)(21,24)(22,25);; 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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2,
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26); s1 := Sym(27)!( 2, 3)( 4, 5)( 7, 9)(10,26)(11,25)(12,27)(13,21)(14,20)(15,19)(16,22)(17,24)(18,23); s2 := Sym(27)!( 1,10)( 2,17)( 3,15)( 4,16)( 5,14)( 6,12)( 7,13)( 8,11)( 9,18)(20,26)(21,24)(22,25); poly := sub<Sym(27)|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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2, s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 >;
References
None.
to this polytope.