Overview
- Group
- SmallGroup(1620,422)
- Rank
- 3
- Schläfli Type
- {6,5}
- Vertices, edges, …
- 162, 405, 135
- Order of s0s1s2
- 10
- 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
81-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*s2*s1*s0*s2*(s1*s0)^2*(s2*s1)^2> of order 3
45 facets
- 45 of {6}*12
54 vertex figures
- 54 of {5}*10
P/N, where N=<s0*(s1*s0*s2)^2*(s1*s0)^2*s2*s1*s0*s2*s1> of order 3
45 facets
- 45 of {6}*12
54 vertex figures
- 54 of {5}*10
P/N, where N=<(s0*s1*s0*(s2*s1)^2)^2> of order 3
45 facets
- 45 of {6}*12
54 vertex figures
- 54 of {5}*10
P/N, where N=<(s1*s0*s2)^2*(s1*s0)^2*(s2*s1)^2> of order 3
45 facets
- 45 of {6}*12
54 vertex figures
- 54 of {5}*10
P/N, where N=<s0*(s1*s0*s2)^2*s1*s0*(s1*s2)^2*s1> of order 3
45 facets
- 45 of {6}*12
54 vertex figures
- 54 of {5}*10
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s1*(s2*s1*s0)^2*s2*s1*s2> of order 3
45 facets
- 45 of {6}*12
54 vertex figures
- 54 of {5}*10
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*s0*(s2*s1)^2*s0*(s1*s2)^2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, (s1*s0*s2)^2*(s1*s0)^2*(s2*s1)^2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2, s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 9
21 facets
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2, s0*s1*s2*s1*s0*s2*(s1*s0)^2*(s2*s1)^2> of order 9
21 facets
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2, s1*(s2*s1*s0)^2*(s1*s2)^2*s1> of order 9
27 facets
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 9
21 facets
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*(s1*s2)^2)^2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9
21 facets
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s1*s0*s1*s2)^2, s0*(s1*s0*s2)^2*(s1*s0)^2*s2*s1*s0*s2*s1> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s2*s1*s0*s2*(s1*s0)^2*(s2*s1)^2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2, s0*s1*s2*s1*s0*(s2*s1)^2*(s0*s1)^2*s2*s1*s2> of order 9
21 facets
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*(s1*s0)^2*(s2*s1)^2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2, s1*s2*s1*s0*s2*(s1*s0)^2*s1*s2*s1*s0*s2*s1*s2> of order 9
21 facets
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s1*s0*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2, s0*s1*(s2*s1*s0)^3*s2*s1*s2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^3*s2*s1*s2> of order 9
21 facets
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9
15 facets
- 15 of {6}*12
18 vertex figures
- 18 of {5}*10
P/N, where N=<(s0*s1)^2, (s2*s1*s0)^2*(s1*s2)^2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 27
9 facets
6 vertex figures
- 6 of {5}*10
P/N, where N=<(s0*s1)^2, s0*(s2*s1)^2*s0*(s1*s2)^2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 27
7 facets
6 vertex figures
- 6 of {5}*10
P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^2, (s2*s1*s0)^2*(s1*s2)^2> of order 27
11 facets
6 vertex figures
- 6 of {5}*10
P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s2*(s1*s0)^2*(s2*s1)^2> of order 27
9 facets
6 vertex figures
- 6 of {5}*10
P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*s1*s2)^2, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 27
5 facets
- 5 of {6}*12
6 vertex figures
- 6 of {5}*10
P/N, where N=<(s0*s1)^2, s0*(s2*s1)^2*s0*(s1*s2)^2, s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 27
7 facets
6 vertex figures
- 6 of {5}*10
P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^2, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 27
11 facets
6 vertex figures
- 6 of {5}*10
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, 2)( 4,71)( 5,70)( 6,72)( 7,50)( 8,49)( 9,51)(10,35)(11,34)(12,36)(13,14)(16,74)(17,73)(18,75)(19,59)(20,58)(21,60)(22,38)(23,37)(24,39)(25,26)(28,80)(29,79)(30,81)(31,32)(40,56)(41,55)(42,57)(43,44)(46,47)(52,68)(53,67)(54,69)(61,62)(64,65)(76,77);; s2 := ( 2,59)( 3,36)( 4,61)( 5,29)( 7,31)( 9,57)(10,26)(11,75)(12,49)(13,77)(14,54)(15,19)(16,47)(17,24)(18,79)(20,70)(21,38)(22,66)(23,40)(25,45)(27,68)(30,63)(32,56)(34,58)(37,53)(39,76)(41,81)(42,46)(43,74)(44,51)(48,65)(50,67)(52,72)(64,80)(69,73)(71,78);; 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*s1*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,
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s2*s1*s2*s1*s2 ];;
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, 2)( 4,71)( 5,70)( 6,72)( 7,50)( 8,49)( 9,51)(10,35)(11,34)(12,36)(13,14)(16,74)(17,73)(18,75)(19,59)(20,58)(21,60)(22,38)(23,37)(24,39)(25,26)(28,80)(29,79)(30,81)(31,32)(40,56)(41,55)(42,57)(43,44)(46,47)(52,68)(53,67)(54,69)(61,62)(64,65)(76,77); s2 := Sym(81)!( 2,59)( 3,36)( 4,61)( 5,29)( 7,31)( 9,57)(10,26)(11,75)(12,49)(13,77)(14,54)(15,19)(16,47)(17,24)(18,79)(20,70)(21,38)(22,66)(23,40)(25,45)(27,68)(30,63)(32,56)(34,58)(37,53)(39,76)(41,81)(42,46)(43,74)(44,51)(48,65)(50,67)(52,72)(64,80)(69,73)(71,78); 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, s1*s2*s1*s2*s1*s2*s1*s2*s1*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, s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s2*s1*s2*s1*s2 >;
References
None.
to this polytope.