Overview
- Group
- SmallGroup(1296,2909)
- Rank
- 3
- Schläfli Type
- {12,4}
- Vertices, edges, …
- 162, 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
6-fold
9-fold
18-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*s1*s2*s1*s0*s1*s2*(s1*s0)^2*s2*s1*s0*s1> of order 2
27 facets
- 27 of {12}*24
81 vertex figures
- 81 of {4}*8
P/N, where N=<(s0*s1)^6> of order 2
30 facets
90 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s2*s1> of order 3
18 facets
- 18 of {12}*24
54 vertex figures
- 54 of {4}*8
P/N, where N=<(s0*s1)^6, (s0*s1)^2*s0*(s2*s1*s0*s1)^2*s2*s1> of order 4
15 facets
45 vertex figures
P/N, where N=<(s0*s1)^3> of order 4
18 facets
45 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 6
12 facets
36 vertex figures
P/N, where N=<(s0*s1)^6, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 6
12 facets
36 vertex figures
P/N, where N=<(s0*s1)^2> of order 6
20 facets
30 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,55)(20,57)(21,56)(22,61)(23,63)(24,62)(25,58)(26,60)(27,59)(37,38)(40,44)(41,43)(42,45)(46,66)(47,65)(48,64)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(73,74)(76,80)(77,79)(78,81);; s1 := ( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,23)(11,22)(12,24)(13,20)(14,19)(15,21)(16,26)(17,25)(18,27)(28,32)(29,31)(30,33)(34,35)(37,50)(38,49)(39,51)(40,47)(41,46)(42,48)(43,53)(44,52)(45,54)(55,59)(56,58)(57,60)(61,62)(64,77)(65,76)(66,78)(67,74)(68,73)(69,75)(70,80)(71,79)(72,81);; s2 := ( 1,39)( 2,38)( 3,37)( 4,42)( 5,41)( 6,40)( 7,45)( 8,44)( 9,43)(10,12)(13,15)(16,18)(19,66)(20,65)(21,64)(22,69)(23,68)(24,67)(25,72)(26,71)(27,70)(28,29)(31,32)(34,35)(46,55)(47,57)(48,56)(49,58)(50,60)(51,59)(52,61)(53,63)(54,62)(73,74)(76,77)(79,80);; 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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,55)(20,57)(21,56)(22,61)(23,63)(24,62)(25,58)(26,60)(27,59)(37,38)(40,44)(41,43)(42,45)(46,66)(47,65)(48,64)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(73,74)(76,80)(77,79)(78,81); s1 := Sym(81)!( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,23)(11,22)(12,24)(13,20)(14,19)(15,21)(16,26)(17,25)(18,27)(28,32)(29,31)(30,33)(34,35)(37,50)(38,49)(39,51)(40,47)(41,46)(42,48)(43,53)(44,52)(45,54)(55,59)(56,58)(57,60)(61,62)(64,77)(65,76)(66,78)(67,74)(68,73)(69,75)(70,80)(71,79)(72,81); s2 := Sym(81)!( 1,39)( 2,38)( 3,37)( 4,42)( 5,41)( 6,40)( 7,45)( 8,44)( 9,43)(10,12)(13,15)(16,18)(19,66)(20,65)(21,64)(22,69)(23,68)(24,67)(25,72)(26,71)(27,70)(28,29)(31,32)(34,35)(46,55)(47,57)(48,56)(49,58)(50,60)(51,59)(52,61)(53,63)(54,62)(73,74)(76,77)(79,80); 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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1 >;
References
None.
to this polytope.