Overview
- Group
- SmallGroup(1296,2908)
- Rank
- 3
- Schläfli Type
- {4,18}
- Vertices, edges, …
- 36, 324, 162
- Order of s0s1s2
- 36
- 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
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=<s2*s1*s0*(s2*s1)^3*s0*(s2*s1)^4*s2> of order 2
81 facets
- 81 of {4}*8
18 vertex figures
- 18 of {18}*36
P/N, where N=<s1*s2*s1*s0*(s2*s1)^3*s0*(s2*s1)^4*s2> of order 2
81 facets
- 81 of {4}*8
21 vertex figures
P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 3
54 facets
- 54 of {4}*8
12 vertex figures
- 12 of {18}*36
P/N, where N=<s0*s1*s0*s2*s1*s0*(s2*s1)^3*s0*s2*s1*s2> of order 3
54 facets
- 54 of {4}*8
12 vertex figures
- 12 of {18}*36
P/N, where N=<s1*s0*(s2*s1)^2*s0*(s2*s1)^3*s2> of order 3
54 facets
- 54 of {4}*8
12 vertex figures
- 12 of {18}*36
P/N, where N=<(s0*s1)^2, s1*s2*s1*s0*(s2*s1)^3*s0*(s2*s1)^4*s2> of order 4
45 facets
12 vertex figures
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*(s1*s2)^2*s1*s0*(s2*s1)^6*s2> of order 6
27 facets
- 27 of {4}*8
9 vertex figures
P/N, where N=<(s0*s1)^2, (s0*s2*s1)^2*s0*s1*s2*s1*s0*s2> of order 6
36 facets
6 vertex figures
- 6 of {18}*36
P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2, s0*s1*s2*s1*s0*(s2*s1)^4*s2> of order 9
18 facets
- 18 of {4}*8
4 vertex figures
- 4 of {18}*36
Representations
Permutation Representation (GAP)
s0 := (10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);; s1 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)(22,26)(23,25)(24,27)(28,55)(29,57)(30,56)(31,62)(32,61)(33,63)(34,59)(35,58)(36,60)(37,64)(38,66)(39,65)(40,71)(41,70)(42,72)(43,68)(44,67)(45,69)(46,73)(47,75)(48,74)(49,80)(50,79)(51,81)(52,77)(53,76)(54,78);; s2 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)(10,31)(11,33)(12,32)(13,28)(14,30)(15,29)(16,35)(17,34)(18,36)(19,49)(20,51)(21,50)(22,46)(23,48)(24,47)(25,53)(26,52)(27,54)(55,67)(56,69)(57,68)(58,64)(59,66)(60,65)(61,71)(62,70)(63,72)(73,76)(74,78)(75,77)(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, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72); s1 := Sym(81)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)(22,26)(23,25)(24,27)(28,55)(29,57)(30,56)(31,62)(32,61)(33,63)(34,59)(35,58)(36,60)(37,64)(38,66)(39,65)(40,71)(41,70)(42,72)(43,68)(44,67)(45,69)(46,73)(47,75)(48,74)(49,80)(50,79)(51,81)(52,77)(53,76)(54,78); s2 := Sym(81)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)(10,31)(11,33)(12,32)(13,28)(14,30)(15,29)(16,35)(17,34)(18,36)(19,49)(20,51)(21,50)(22,46)(23,48)(24,47)(25,53)(26,52)(27,54)(55,67)(56,69)(57,68)(58,64)(59,66)(60,65)(61,71)(62,70)(63,72)(73,76)(74,78)(75,77)(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, s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References
None.
to this polytope.