Overview
- Group
- SmallGroup(1944,2344)
- Rank
- 3
- Schläfli Type
- {6,6}
- Vertices, edges, …
- 162, 486, 162
- Order of s0s1s2
- 18
- 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
3-fold
6-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*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<(s0*s1)^2*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2*s1> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<s1*(s2*s1*s0)^4*s2*s1*s2> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<s0*s2*s1*s0*(s2*s1)^2*s0*s1*s2*s1*s0*s2*s1*s2> of order 3
54 facets
- 54 of {6}*12
54 vertex figures
- 54 of {6}*12
P/N, where N=<(s1*s0)^2*(s2*s1)^2*(s0*s1)^2*(s2*s1)^2> of order 3
54 facets
- 54 of {6}*12
60 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*(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=<s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 3
54 facets
- 54 of {6}*12
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*s2*s1*s0*(s2*s1)^2*(s0*s2*s1)^2*s2, s0*s2*(s1*s0)^2*(s2*s1)^2*(s0*s1)^2*s2*s1*s2> of order 6
27 facets
- 27 of {6}*12
27 vertex figures
- 27 of {6}*12
P/N, where N=<(s1*s0)^2*s1*s2*s1*s0*s2*s1*s2> of order 6
27 facets
- 27 of {6}*12
27 vertex figures
- 27 of {6}*12
P/N, where N=<(s0*s1)^3, s0*(s2*s1)^2*s0*s2*s1*s0*(s2*s1)^2*s2> of order 6
30 facets
27 vertex figures
- 27 of {6}*12
P/N, where N=<(s0*s1)^3, s1*(s2*s1*s0)^4*s2*s1*s2> of order 6
30 facets
27 vertex figures
- 27 of {6}*12
P/N, where N=<(s0*s1)^3, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 6
36 facets
27 vertex figures
- 27 of {6}*12
P/N, where N=<(s0*s1)^3, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 6
30 facets
27 vertex figures
- 27 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*(s0*s2*s1)^2*s2, s2*(s1*s0)^2*(s2*s1)^2*(s0*s1)^2*s2*s1*s2> of order 6
27 facets
- 27 of {6}*12
30 vertex figures
P/N, where N=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1)^2*s0*s2*s1*s0> of order 6
27 facets
- 27 of {6}*12
30 vertex figures
P/N, where N=<(s0*s1)^3*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 6
27 facets
- 27 of {6}*12
30 vertex figures
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*s2*s1, s0*s1*s0*s2*s1*s0*(s2*s1)^2*s0*s1> of order 6
27 facets
- 27 of {6}*12
30 vertex figures
P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s1*(s2*s1*s0)^2*s2> of order 9
18 facets
- 18 of {6}*12
22 vertex figures
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
18 vertex figures
- 18 of {6}*12
P/N, where N=<s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1, (s0*s1)^2*s0*(s2*s1)^2*s0*(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, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9
24 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2, s0*(s1*s0*s2)^2*s1*s0*(s1*s2)^2*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*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> 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*(s1*s2)^2*s1, s1*s2*s1*s0*(s1*s2)^2*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*s1*s2*s1*s0*s2*(s1*s0)^2*(s2*s1)^2, (s0*s1)^2*s2*s1*s0*(s1*s2)^2*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=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9
30 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s1*s0*s1*s2)^2, s1*s0*s2*s1*s0*s1*s2*s1> of order 9
24 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s2*s1*s0)^2*(s1*s2)^2, s1*(s2*s1*s0)^2*(s1*s2)^2*s1> of order 9
30 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, (s1*s2)^2*s1*s0*s2*s1*s0*(s2*s1)^2*s2> of order 9
24 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*s1*s2)^2> of order 9
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*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*s1*s2)^2, s1*(s2*s1*s0)^2*(s1*s2)^2*s1> of order 9
24 facets
18 vertex figures
- 18 of {6}*12
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
18 facets
- 18 of {6}*12
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s0*s1)^2, s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 9
24 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s0*s1)^3, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 18
12 facets
9 vertex figures
- 9 of {6}*12
P/N, where N=<(s0*s1)^3, s0*s1*s0*(s2*s1)^2*s0*(s1*s2)^2> of order 18
12 facets
9 vertex figures
- 9 of {6}*12
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*(s1*s0)^2*s2*s1, s1*s0*(s1*s2)^2*s1*s0*s2*s1> of order 18
9 facets
- 9 of {6}*12
12 vertex figures
P/N, where N=<(s0*s1)^3, s0*s1*(s2*s1*s0)^2*(s1*s2)^2> of order 18
12 facets
9 vertex figures
- 9 of {6}*12
P/N, where N=<(s1*s2)^3, s0*(s1*s2)^2*s1*s0*s2, s1*s0*(s1*s2)^2*s1*s0*s2*s1> of order 18
9 facets
- 9 of {6}*12
12 vertex figures
P/N, where N=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*s1)^3*s2*s1*s0*s2*s1*s2, (s0*s1)^2*(s2*s1)^2*s0*s2*s1*s0> of order 18
9 facets
- 9 of {6}*12
12 vertex figures
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
8 facets
6 vertex figures
- 6 of {6}*12
P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2, (s0*s1*s2*s1)^2> of order 27
12 facets
6 vertex figures
- 6 of {6}*12
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 := ( 1,37)( 2,39)( 3,38)( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,46)(20,48)(21,47)(22,49)(23,51)(24,50)(25,52)(26,54)(27,53)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(74,75)(77,78)(80,81);; s2 := ( 4, 8)( 5, 9)( 6, 7)(10,14)(11,15)(12,13)(19,27)(20,25)(21,26)(28,55)(29,56)(30,57)(31,62)(32,63)(33,61)(34,60)(35,58)(36,59)(37,68)(38,69)(39,67)(40,66)(41,64)(42,65)(43,70)(44,71)(45,72)(46,81)(47,79)(48,80)(49,76)(50,77)(51,78)(52,74)(53,75)(54,73);; 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,
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1 ];;
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)!( 1,37)( 2,39)( 3,38)( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,46)(20,48)(21,47)(22,49)(23,51)(24,50)(25,52)(26,54)(27,53)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(74,75)(77,78)(80,81); s2 := Sym(81)!( 4, 8)( 5, 9)( 6, 7)(10,14)(11,15)(12,13)(19,27)(20,25)(21,26)(28,55)(29,56)(30,57)(31,62)(32,63)(33,61)(34,60)(35,58)(36,59)(37,68)(38,69)(39,67)(40,66)(41,64)(42,65)(43,70)(44,71)(45,72)(46,81)(47,79)(48,80)(49,76)(50,77)(51,78)(52,74)(53,75)(54,73); 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*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1 >;
References
None.
to this polytope.