Overview
- Group
- SmallGroup(1944,2345)
- Rank
- 3
- Schläfli Type
- {6,18}
- Vertices, edges, …
- 54, 486, 162
- Order of s0s1s2
- 18
- 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=<(s1*s0)^2*s2*s1*s0*(s1*s2)^4> of order 2
81 facets
- 81 of {6}*12
27 vertex figures
- 27 of {18}*36
P/N, where N=<(s0*s1)^2*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2*s1> of order 3
54 facets
- 54 of {6}*12
18 vertex figures
- 18 of {18}*36
P/N, where N=<(s1*s0)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2> of order 3
54 facets
- 54 of {6}*12
18 vertex figures
- 18 of {18}*36
P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 3
54 facets
- 54 of {6}*12
18 vertex figures
- 18 of {18}*36
P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, (s0*s1)^2*s0*s2*s1*s0*(s1*s2)^4> of order 6
27 facets
- 27 of {6}*12
12 vertex figures
P/N, where N=<(s1*s0)^2*(s2*s1)^2*s0*s1*s0*s2*s1*s2, (s0*s1)^2*s0*s2*s1*s0*(s1*s2)^4> of order 6
27 facets
- 27 of {6}*12
12 vertex figures
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)(10,52)(11,54)(12,53)(13,48)(14,47)(15,46)(16,50)(17,49)(18,51)(19,42)(20,41)(21,40)(22,44)(23,43)(24,45)(25,37)(26,39)(27,38)(56,57)(58,60)(61,62)(64,79)(65,81)(66,80)(67,75)(68,74)(69,73)(70,77)(71,76)(72,78);; s2 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(19,20)(22,26)(23,25)(24,27)(28,64)(29,66)(30,65)(31,70)(32,72)(33,71)(34,67)(35,69)(36,68)(37,55)(38,57)(39,56)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,74)(47,73)(48,75)(49,80)(50,79)(51,81)(52,77)(53,76)(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, s0*s1*s0*s1*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*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s2*s0*s1*s2*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)(10,52)(11,54)(12,53)(13,48)(14,47)(15,46)(16,50)(17,49)(18,51)(19,42)(20,41)(21,40)(22,44)(23,43)(24,45)(25,37)(26,39)(27,38)(56,57)(58,60)(61,62)(64,79)(65,81)(66,80)(67,75)(68,74)(69,73)(70,77)(71,76)(72,78); s2 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(19,20)(22,26)(23,25)(24,27)(28,64)(29,66)(30,65)(31,70)(32,72)(33,71)(34,67)(35,69)(36,68)(37,55)(38,57)(39,56)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,74)(47,73)(48,75)(49,80)(50,79)(51,81)(52,77)(53,76)(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, s0*s1*s0*s1*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*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1 >;
References
None.
to this polytope.