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
36-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*s1*s0*(s2*(s1*s0)^2)^2*s2*s1> of order 2
27 facets
- 27 of {12}*24
54 vertex figures
- 54 of {6}*12
P/N, where N=<s2*s1*s0*(s1*s2)^2*s1*s0*s2*s1*s2> of order 2
27 facets
- 27 of {12}*24
57 vertex figures
P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*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*(s2*s1)^2*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*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, s2*s1*s0*(s1*s2)^2*s1*s0*s2*s1*s2> of order 4
15 facets
30 vertex figures
P/N, where N=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s1> of order 6
9 facets
- 9 of {12}*24
21 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, (s0*s1)^6> of order 6
12 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<(s0*s1)^6, (s0*s1)^2*s2*(s1*s0)^2*s1*s2*s1> of order 6
12 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2, (s1*s0*s1*s2)^2> of order 9
6 facets
- 6 of {12}*24
18 vertex figures
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1*s0)^2> of order 9
6 facets
- 6 of {12}*24
24 vertex figures
P/N, where N=<(s0*s1)^4, (s0*s1)^2*s2*(s1*s0)^2*s2> of order 9
12 facets
12 vertex figures
- 12 of {6}*12
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);; s1 := ( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,32)(11,31)(12,33)(13,29)(14,28)(15,30)(16,35)(17,34)(18,36)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)(27,63)(37,41)(38,40)(39,42)(43,44)(46,68)(47,67)(48,69)(49,65)(50,64)(51,66)(52,71)(53,70)(54,72)(73,77)(74,76)(75,78)(79,80);; s2 := ( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)(22,25)(23,26)(24,27)(28,64)(29,65)(30,66)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)(37,55)(38,56)(39,57)(40,61)(41,62)(42,63)(43,58)(44,59)(45,60)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78);; 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*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*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)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77); s1 := Sym(81)!( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,32)(11,31)(12,33)(13,29)(14,28)(15,30)(16,35)(17,34)(18,36)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)(27,63)(37,41)(38,40)(39,42)(43,44)(46,68)(47,67)(48,69)(49,65)(50,64)(51,66)(52,71)(53,70)(54,72)(73,77)(74,76)(75,78)(79,80); s2 := Sym(81)!( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)(22,25)(23,26)(24,27)(28,64)(29,65)(30,66)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)(37,55)(38,56)(39,57)(40,61)(41,62)(42,63)(43,58)(44,59)(45,60)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78); 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*s1*s2, s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*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.