Overview
- Group
- SmallGroup(1296,3509)
- Rank
- 3
- Schläfli Type
- {6,8}
- Vertices, edges, …
- 81, 324, 108
- Order of s0s1s2
- 8
- Order of s0s1s2s1
- 6
- Also known as
- if this polytope has a name.
Special Properties
- Compact Hyperbolic Quotient
- Locally Spherical
- Non-Orientable
Quotients maximal quotients in bold
No regular quotients.
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*s0*s2*(s1*s0)^2*s2*s1*s2> of order 3
36 facets
- 36 of {6}*12
27 vertex figures
- 27 of {8}*16
P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2> of order 3
36 facets
- 36 of {6}*12
27 vertex figures
- 27 of {8}*16
P/N, where N=<s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 3
36 facets
- 36 of {6}*12
27 vertex figures
- 27 of {8}*16
P/N, where N=<(s0*s1)^2*s2*s1*s0*(s2*s1)^3> of order 3
36 facets
- 36 of {6}*12
27 vertex figures
- 27 of {8}*16
P/N, where N=<(s0*s1)^3*(s2*s1*s0)^2*s2*s1*s2> of order 3
36 facets
- 36 of {6}*12
27 vertex figures
- 27 of {8}*16
P/N, where N=<s0*s2*s1*s0*s1*s2, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 9
24 facets
9 vertex figures
- 9 of {8}*16
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, (s2*s1*s0)^2*(s1*s2)^2> of order 9
18 facets
9 vertex figures
- 9 of {8}*16
P/N, where N=<(s2*s1*s0)^2*(s1*s2)^2, s0*s1*s0*s2*s1*s0*(s2*s1)^2*s2> of order 9
18 facets
9 vertex figures
- 9 of {8}*16
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 9
12 facets
- 12 of {6}*12
9 vertex figures
- 9 of {8}*16
P/N, where N=<(s0*s1)^2*s2*s1*s0*(s2*s1)^3, s0*s1*s2*(s1*s0)^2*(s1*s2)^3> of order 9
12 facets
- 12 of {6}*12
9 vertex figures
- 9 of {8}*16
P/N, where N=<s0*s1*s0*s2*s1*s0*(s2*s1)^2*s2, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9
12 facets
- 12 of {6}*12
9 vertex figures
- 9 of {8}*16
P/N, where N=<(s2*s1*s0)^2*(s1*s2)^2, (s0*s1)^2*s2*s1*s0*(s2*s1)^3> of order 9
18 facets
9 vertex figures
- 9 of {8}*16
P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2, s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1> of order 9
12 facets
- 12 of {6}*12
9 vertex figures
- 9 of {8}*16
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,73)(38,75)(39,74)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,64)(47,66)(48,65)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68);; s1 := ( 1,28)( 2,29)( 3,30)( 4,37)( 5,38)( 6,39)( 7,46)( 8,47)( 9,48)(10,31)(11,32)(12,33)(13,40)(14,41)(15,42)(16,49)(17,50)(18,51)(19,34)(20,35)(21,36)(22,43)(23,44)(24,45)(25,52)(26,53)(27,54)(58,64)(59,65)(60,66)(61,73)(62,74)(63,75)(70,76)(71,77)(72,78);; s2 := ( 2,10)( 3,19)( 4,55)( 5,64)( 6,73)( 7,28)( 8,37)( 9,46)(12,20)(13,56)(14,65)(15,74)(16,29)(17,38)(18,47)(22,57)(23,66)(24,75)(25,30)(26,39)(27,48)(31,61)(32,70)(33,79)(35,43)(36,52)(40,62)(41,71)(42,80)(45,53)(49,63)(50,72)(51,81)(59,67)(60,76)(69,77);; 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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1,
s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,73)(38,75)(39,74)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,64)(47,66)(48,65)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68); s1 := Sym(81)!( 1,28)( 2,29)( 3,30)( 4,37)( 5,38)( 6,39)( 7,46)( 8,47)( 9,48)(10,31)(11,32)(12,33)(13,40)(14,41)(15,42)(16,49)(17,50)(18,51)(19,34)(20,35)(21,36)(22,43)(23,44)(24,45)(25,52)(26,53)(27,54)(58,64)(59,65)(60,66)(61,73)(62,74)(63,75)(70,76)(71,77)(72,78); s2 := Sym(81)!( 2,10)( 3,19)( 4,55)( 5,64)( 6,73)( 7,28)( 8,37)( 9,46)(12,20)(13,56)(14,65)(15,74)(16,29)(17,38)(18,47)(22,57)(23,66)(24,75)(25,30)(26,39)(27,48)(31,61)(32,70)(33,79)(35,43)(36,52)(40,62)(41,71)(42,80)(45,53)(49,63)(50,72)(51,81)(59,67)(60,76)(69,77); 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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1, s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0 >;
References
None.
to this polytope.