Overview
- Group
- SmallGroup(1620,422)
- Rank
- 3
- Schläfli Type
- {5,6}
- Vertices, edges, …
- 135, 405, 162
- 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=<(s1*s0)^2*(s2*s1)^2*(s0*s2*s1)^2> of order 3
54 facets
- 54 of {5}*10
45 vertex figures
- 45 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^2*(s1*s0*s2)^2*s1*s2> of order 3
54 facets
- 54 of {5}*10
45 vertex figures
- 45 of {6}*12
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1*s0*(s1*s2)^2*s1> of order 3
54 facets
- 54 of {5}*10
45 vertex figures
- 45 of {6}*12
P/N, where N=<(s1*s0)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2> of order 3
54 facets
- 54 of {5}*10
45 vertex figures
- 45 of {6}*12
P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s2> of order 3
54 facets
- 54 of {5}*10
45 vertex figures
- 45 of {6}*12
P/N, where N=<s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s2*s1> of order 3
54 facets
- 54 of {5}*10
45 vertex figures
- 45 of {6}*12
P/N, where N=<(s0*s1)^2*(s0*s2*s1)^3*s2> of order 3
54 facets
- 54 of {5}*10
45 vertex figures
- 45 of {6}*12
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 {5}*10
27 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*s2, (s1*s0)^2*(s2*s1)^2*(s0*s2*s1)^2> of order 9
18 facets
- 18 of {5}*10
21 vertex figures
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s2*s1> of order 9
18 facets
- 18 of {5}*10
21 vertex figures
P/N, where N=<(s1*s2)^2, (s0*s1)^2*(s2*s1*s0)^2> of order 9
18 facets
- 18 of {5}*10
27 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, (s1*s0)^2*(s2*s1)^2*s0*s2*s1*s0*s1*s2> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2, (s1*s0)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2> of order 9
18 facets
- 18 of {5}*10
21 vertex figures
P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s2*s1> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s2> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s2, s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*s1*s0*s2*(s1*s0)^2*s2*s1*s0*s2*s1*s2> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1*s0*s1, s0*(s1*s0*s2)^4*s1> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<(s0*s1)^2*(s0*s2*s1)^3*s2, (s0*s1)^2*(s2*s1*s0)^2*(s2*s1)^2> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1*s0*s1, (s0*s1)^2*(s0*s2*s1)^3*s2> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^3*s1*s0*s1*s2, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s1*s2*s1> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<(s0*s1)^2*(s0*s2*s1)^2*s0*s1, (s0*s1)^2*(s2*s1*s0)^2*(s2*s1)^2> of order 9
18 facets
- 18 of {5}*10
21 vertex figures
P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s1*s2*s1, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2, s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s1*s2> of order 9
18 facets
- 18 of {5}*10
21 vertex figures
P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1)^2*(s2*s1*s0)^2> of order 9
18 facets
- 18 of {5}*10
21 vertex figures
P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*(s2*s1)^2*s2, s0*(s1*s0*s2)^4*s1> of order 9
18 facets
- 18 of {5}*10
15 vertex figures
- 15 of {6}*12
P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*(s2*s1*s0)^2> of order 9
18 facets
- 18 of {5}*10
21 vertex figures
P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*s1*s2)^2, (s0*s1)^2*(s2*s1*s0)^2> of order 27
6 facets
- 6 of {5}*10
7 vertex figures
P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*(s2*s1)^2*s0*s2*s1*s2, (s0*s1)^2*(s0*s2*s1)^2*s0*s1> of order 27
6 facets
- 6 of {5}*10
7 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*s2, (s1*s0*s1*s2)^2, (s0*s1)^2*(s0*s2*s1)^3*s2> of order 27
6 facets
- 6 of {5}*10
7 vertex figures
P/N, where N=<(s1*s2)^2, (s0*s1)^2*(s2*s1*s0)^2, (s0*s1)^2*(s0*s2*s1)^2*s0*s1> of order 27
6 facets
- 6 of {5}*10
11 vertex figures
P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*s1*s2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1*s0)^2> of order 27
6 facets
- 6 of {5}*10
11 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, s1*s0*(s2*s1)^2*s0*s2*s1*s2, (s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1> of order 27
6 facets
- 6 of {5}*10
5 vertex figures
- 5 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2, (s0*s1)^2*s2*(s1*s0)^2*s2, s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 27
6 facets
- 6 of {5}*10
9 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 4,49)( 5,50)( 6,51)( 7,70)( 8,71)( 9,72)(10,58)(11,59)(12,60)(13,25)(14,26)(15,27)(16,37)(17,38)(18,39)(19,34)(20,35)(21,36)(22,73)(23,74)(24,75)(28,40)(29,41)(30,42)(31,61)(32,62)(33,63)(43,76)(44,77)(45,78)(46,64)(47,65)(48,66)(55,79)(56,80)(57,81);; s1 := ( 2,36)( 3,59)( 4,31)( 5,57)( 6, 8)( 7,61)( 9,29)(10,15)(11,38)(12,70)(13,45)(14,68)(16,66)(18,40)(19,26)(20,49)(21,75)(22,47)(23,79)(25,77)(27,54)(28,55)(30,32)(33,62)(35,60)(37,69)(39,43)(42,64)(44,71)(46,80)(50,52)(51,78)(53,73)(56,63)(67,72)(74,76);; s2 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(28,56)(29,55)(30,57)(31,62)(32,61)(33,63)(34,59)(35,58)(36,60)(37,74)(38,73)(39,75)(40,80)(41,79)(42,81)(43,77)(44,76)(45,78)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69);; 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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!( 4,49)( 5,50)( 6,51)( 7,70)( 8,71)( 9,72)(10,58)(11,59)(12,60)(13,25)(14,26)(15,27)(16,37)(17,38)(18,39)(19,34)(20,35)(21,36)(22,73)(23,74)(24,75)(28,40)(29,41)(30,42)(31,61)(32,62)(33,63)(43,76)(44,77)(45,78)(46,64)(47,65)(48,66)(55,79)(56,80)(57,81); s1 := Sym(81)!( 2,36)( 3,59)( 4,31)( 5,57)( 6, 8)( 7,61)( 9,29)(10,15)(11,38)(12,70)(13,45)(14,68)(16,66)(18,40)(19,26)(20,49)(21,75)(22,47)(23,79)(25,77)(27,54)(28,55)(30,32)(33,62)(35,60)(37,69)(39,43)(42,64)(44,71)(46,80)(50,52)(51,78)(53,73)(56,63)(67,72)(74,76); s2 := Sym(81)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(28,56)(29,55)(30,57)(31,62)(32,61)(33,63)(34,59)(35,58)(36,60)(37,74)(38,73)(39,75)(40,80)(41,79)(42,81)(43,77)(44,76)(45,78)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69); 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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 >;
References
None.
to this polytope.