Overview
- Group
- SmallGroup(1296,3529)
- Rank
- 3
- Schläfli Type
- {12,12}
- Vertices, edges, …
- 54, 324, 54
- Order of s0s1s2
- 6
- 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
9-fold
18-fold
36-fold
54-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)^2*s0*s1*s2*s1*s0*(s1*s2)^2> of order 2
27 facets
- 27 of {12}*24
27 vertex figures
- 27 of {12}*24
P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*(s2*s1)^2*s0*s2> of order 2
27 facets
- 27 of {12}*24
27 vertex figures
- 27 of {12}*24
P/N, where N=<(s0*s1)^6> of order 2
30 facets
30 vertex figures
P/N, where N=<s1*s0*s1*s2*s1*s0*(s2*s1)^3> of order 3
18 facets
- 18 of {12}*24
18 vertex figures
- 18 of {12}*24
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2> of order 3
18 facets
- 18 of {12}*24
18 vertex figures
- 18 of {12}*24
P/N, where N=<s0*s1*s2*(s1*s0)^2*s2*s1*s0*(s2*s1)^2> of order 3
18 facets
- 18 of {12}*24
18 vertex figures
- 18 of {12}*24
P/N, where N=<(s0*s1)^6, s2*(s1*s0)^2*s2*s1*s0*(s2*s1)^2*s2> of order 4
15 facets
15 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)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)(20,24)(21,23)(26,27)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,43)(35,45)(36,44)(46,49)(47,51)(48,50)(53,54)(55,67)(56,69)(57,68)(58,64)(59,66)(60,65)(61,70)(62,72)(63,71)(73,76)(74,78)(75,77)(80,81);; 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, s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*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)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)(20,24)(21,23)(26,27)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,43)(35,45)(36,44)(46,49)(47,51)(48,50)(53,54)(55,67)(56,69)(57,68)(58,64)(59,66)(60,65)(61,70)(62,72)(63,71)(73,76)(74,78)(75,77)(80,81); 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, s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*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.