Overview
- Group
- SmallGroup(1944,2344)
- Rank
- 3
- Schläfli Type
- {6,18}
- Vertices, edges, …
- 54, 486, 162
- 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
27-fold
54-fold
81-fold
162-fold
243-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)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 3
54 facets
- 54 of {6}*12
18 vertex figures
- 18 of {18}*36
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1> of order 3
54 facets
- 54 of {6}*12
18 vertex figures
- 18 of {18}*36
P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2> of order 3
54 facets
- 54 of {6}*12
18 vertex figures
- 18 of {18}*36
P/N, where N=<((s1*s0)^2*s1*s2)^2> of order 3
54 facets
- 54 of {6}*12
18 vertex figures
- 18 of {18}*36
P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 3
54 facets
- 54 of {6}*12
18 vertex figures
- 18 of {18}*36
P/N, where N=<s1*s0*(s1*s2)^3*s1*s0*s1*s2> of order 3
54 facets
- 54 of {6}*12
18 vertex figures
- 18 of {18}*36
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, (s0*(s2*s1)^2)^2> of order 9
18 facets
- 18 of {6}*12
6 vertex figures
- 6 of {18}*36
P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1, (s0*s1)^3*s2*s1*s0*s1*s2*s1> of order 9
18 facets
- 18 of {6}*12
6 vertex figures
- 6 of {18}*36
Representations
Permutation Representation (GAP)
s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)(40,70)(41,71)(42,72)(43,67)(44,68)(45,69)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78);; s1 := ( 1,28)( 2,30)( 3,29)( 4,33)( 5,32)( 6,31)( 7,35)( 8,34)( 9,36)(10,54)(11,53)(12,52)(13,47)(14,46)(15,48)(16,49)(17,51)(18,50)(19,41)(20,40)(21,42)(22,43)(23,45)(24,44)(25,39)(26,38)(27,37)(56,57)(58,60)(61,62)(64,81)(65,80)(66,79)(67,74)(68,73)(69,75)(70,76)(71,78)(72,77);; s2 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)(36,44)(47,48)(50,51)(53,54)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(74,75)(77,78)(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*s0*s1*s0*s1*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*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)(40,70)(41,71)(42,72)(43,67)(44,68)(45,69)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78); s1 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,33)( 5,32)( 6,31)( 7,35)( 8,34)( 9,36)(10,54)(11,53)(12,52)(13,47)(14,46)(15,48)(16,49)(17,51)(18,50)(19,41)(20,40)(21,42)(22,43)(23,45)(24,44)(25,39)(26,38)(27,37)(56,57)(58,60)(61,62)(64,81)(65,80)(66,79)(67,74)(68,73)(69,75)(70,76)(71,78)(72,77); s2 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)(36,44)(47,48)(50,51)(53,54)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(74,75)(77,78)(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*s0*s1*s0*s1*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*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 >;
References
None.
to this polytope.