Overview
- Group
- SmallGroup(1944,2342)
- Rank
- 3
- Schläfli Type
- {6,6}
- Vertices, edges, …
- 162, 486, 162
- Order of s0s1s2
- 6
- Order of s0s1s2s1
- 6
- Also known as
- if this polytope has a name.
Special Properties
- Compact Hyperbolic Quotient
- Locally Spherical
- Orientable
- Self-Petrie
Quotients maximal quotients in bold
3-fold
9-fold
18-fold
27-fold
54-fold
81-fold
162-fold
243-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)^3*(s2*(s1*s0)^2*s1)^2*s2> of order 2
81 facets
- 81 of {6}*12
81 vertex figures
- 81 of {6}*12
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^2*(s1*s0)^2*s2*s1*s0*s2*s1> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<(s0*s1)^3*s2*(s1*s0)^2*s2*s1*s0*(s1*s2)^2> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 3
60 facets
54 vertex figures
- 54 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^2*s1*s2, (s0*s2*s1)^3> of order 6
27 facets
- 27 of {6}*12
27 vertex figures
- 27 of {6}*12
P/N, where N=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, s2*s1*s0*(s1*s2)^2*s1*s0*s2*s1*s2> of order 6
27 facets
- 27 of {6}*12
30 vertex figures
P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, s1*s0*s2*(s1*s0)^2*s2*s1*s0*(s1*s2)^3> of order 6
27 facets
- 27 of {6}*12
30 vertex figures
P/N, where N=<(s1*s2)^2, (s0*s1)^2*(s2*s1*s0)^2> of order 9
18 facets
- 18 of {6}*12
36 vertex figures
P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*(s2*s1)^2*s2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s1*s0)^2*(s2*s1*s0*s1)^2, (s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 9
24 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s0*s1*s2*s1)^2, ((s1*s0)^2*s1*s2)^2> of order 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9
18 facets
- 18 of {6}*12
24 vertex figures
P/N, where N=<((s1*s0)^2*s1*s2)^2, s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s1*s2> of order 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s1*s0*s1*s2)^2, (s1*s0)^2*s1*s2*(s1*s0)^2*s2*s1> of order 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1*s0)^2> of order 9
18 facets
- 18 of {6}*12
30 vertex figures
P/N, where N=<(s1*s0*s1*s2)^2, ((s1*s0)^2*s1*s2)^2> of order 9
18 facets
- 18 of {6}*12
24 vertex figures
P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, ((s1*s0)^2*s1*s2)^2> of order 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, (s1*s0)^2*s1*(s2*s1*s0)^2*s1> of order 9
18 facets
- 18 of {6}*12
24 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*s2, (s0*s1)^2*s2*(s1*s0)^2*s2> of order 9
18 facets
- 18 of {6}*12
30 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, (s1*s0)^2*s1*s2*(s1*s0)^2*s2*s1> of order 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, s0*(s1*s0*s2)^2*(s1*s0)^2*s2*s1*s0*s2*s1> of order 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1)^2*(s2*s1*s0)^2> of order 9
18 facets
- 18 of {6}*12
24 vertex figures
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 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1, (s0*s1)^3*s2*s1*s0*s1*s2*s1> of order 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s0*s1)^3, s0*(s1*s0*s2)^2*s1> of order 18
12 facets
15 vertex figures
P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1, s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*s1)^3*s2*s1*s0*s1*s2*s1> of order 18
9 facets
- 9 of {6}*12
12 vertex figures
P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*s2*(s1*s0)^2*s2, s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 27
6 facets
- 6 of {6}*12
8 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26);; s1 := ( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,10)( 8,11)( 9,12);; s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,11)(13,17)(14,16)(15,18)(19,21)(22,27)(23,26)(24,25);; 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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(27)!( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26); s1 := Sym(27)!( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,10)( 8,11)( 9,12); s2 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,11)(13,17)(14,16)(15,18)(19,21)(22,27)(23,26)(24,25); 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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1 >;
References
None.
to this polytope.