Overview
- Group
- SmallGroup(1296,3529)
- Rank
- 3
- Schläfli Type
- {12,6}
- Vertices, edges, …
- 108, 324, 54
- Order of s0s1s2
- 12
- 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
108-fold
162-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*s2*s1*s0*(s2*(s1*s0)^2)^2*s2*s1*s2> of order 2
27 facets
- 27 of {12}*24
54 vertex figures
- 54 of {6}*12
P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s1*s2*s1> of order 3
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 3
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s2*s1> of order 3
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<(s0*s2*s1)^2*s0*(s1*s2)^2*s1*s0*s1> of order 3
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^3*s1*s2> of order 3
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1> of order 3
18 facets
- 18 of {12}*24
36 vertex figures
- 36 of {6}*12
P/N, where N=<(s0*s1)^6, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 6
12 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 9
6 facets
- 6 of {12}*24
18 vertex figures
P/N, where N=<(s1*s0*s1*s2)^2, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9
6 facets
- 6 of {12}*24
12 vertex figures
- 12 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, s0*s2*(s1*s0)^2*s2*s1*s0*s1> of order 9
6 facets
- 6 of {12}*24
12 vertex figures
- 12 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2, (s0*s1)^2*s0*s2*s1*s0*(s1*s2)^3> of order 9
6 facets
- 6 of {12}*24
18 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,55)(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,64)(38,66)(39,65)(40,67)(41,69)(42,68)(43,70)(44,72)(45,71)(46,73)(47,75)(48,74)(49,76)(50,78)(51,77)(52,79)(53,81)(54,80);; s1 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,29)(11,28)(12,30)(13,35)(14,34)(15,36)(16,32)(17,31)(18,33)(19,56)(20,55)(21,57)(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(37,38)(40,44)(41,43)(42,45)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69)(73,74)(76,80)(77,79)(78,81);; s2 := ( 1,13)( 2,14)( 3,15)( 4,10)( 5,11)( 6,12)( 7,16)( 8,17)( 9,18)(19,22)(20,23)(21,24)(28,67)(29,68)(30,69)(31,64)(32,65)(33,66)(34,70)(35,71)(36,72)(37,58)(38,59)(39,60)(40,55)(41,56)(42,57)(43,61)(44,62)(45,63)(46,76)(47,77)(48,78)(49,73)(50,74)(51,75)(52,79)(53,80)(54,81);; 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*s1*s2,
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,55)(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,64)(38,66)(39,65)(40,67)(41,69)(42,68)(43,70)(44,72)(45,71)(46,73)(47,75)(48,74)(49,76)(50,78)(51,77)(52,79)(53,81)(54,80); s1 := Sym(81)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,29)(11,28)(12,30)(13,35)(14,34)(15,36)(16,32)(17,31)(18,33)(19,56)(20,55)(21,57)(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(37,38)(40,44)(41,43)(42,45)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69)(73,74)(76,80)(77,79)(78,81); s2 := Sym(81)!( 1,13)( 2,14)( 3,15)( 4,10)( 5,11)( 6,12)( 7,16)( 8,17)( 9,18)(19,22)(20,23)(21,24)(28,67)(29,68)(30,69)(31,64)(32,65)(33,66)(34,70)(35,71)(36,72)(37,58)(38,59)(39,60)(40,55)(41,56)(42,57)(43,61)(44,62)(45,63)(46,76)(47,77)(48,78)(49,73)(50,74)(51,75)(52,79)(53,80)(54,81); 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*s1*s2, s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References
None.
to this polytope.