Overview
- Group
- SmallGroup(1296,3509)
- Rank
- 3
- Schläfli Type
- {8,6}
- Vertices, edges, …
- 108, 324, 81
- 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
- Self-Petrie
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=<(s1*s0)^2*s1*s2*(s1*s0)^2*s2*s1> of order 3
27 facets
- 27 of {8}*16
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*(s1*s2)^2*(s1*s0)^3*s2*s1> of order 3
27 facets
- 27 of {8}*16
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 3
27 facets
- 27 of {8}*16
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 3
27 facets
- 27 of {8}*16
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*(s1*s2)^2*(s1*s0)^2*s2*s1*s0*s2*s1*s2> of order 3
27 facets
- 27 of {8}*16
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s2*s1*s2> of order 3
27 facets
- 27 of {8}*16
36 vertex figures
- 36 of {6}*12
P/N, where N=<(s1*s0*s1*s2)^2, (s1*s0)^2*s1*s2*(s1*s0)^2*s2*s1> of order 9
9 facets
- 9 of {8}*16
12 vertex figures
- 12 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^2*s1*s0*s1*s2, (s1*s0)^2*s1*s2*(s1*s0)^2*s2*s1> of order 9
9 facets
- 9 of {8}*16
12 vertex figures
- 12 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2, (s0*s1)^2*(s0*s2*s1)^2*s2> of order 9
9 facets
- 9 of {8}*16
18 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 9
9 facets
- 9 of {8}*16
24 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, (s0*s1)^2*(s0*s2*s1)^2*s2> of order 9
9 facets
- 9 of {8}*16
12 vertex figures
- 12 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, s0*(s1*s0*s2)^3*s1> of order 9
9 facets
- 9 of {8}*16
12 vertex figures
- 12 of {6}*12
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*(s1*s2)^2*s1, (s0*s1)^2*(s2*s1)^2*(s0*s1)^2*s2> of order 9
9 facets
- 9 of {8}*16
12 vertex figures
- 12 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2, (s0*s1)^3*(s2*s1)^2*s0*s1*s2> of order 9
9 facets
- 9 of {8}*16
18 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)(28,55)(29,56)(30,57)(31,64)(32,65)(33,66)(34,73)(35,74)(36,75)(37,58)(38,59)(39,60)(40,67)(41,68)(42,69)(43,76)(44,77)(45,78)(46,61)(47,62)(48,63)(49,70)(50,71)(51,72)(52,79)(53,80)(54,81);; s1 := ( 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);; s2 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(28,56)(29,55)(30,57)(31,62)(32,61)(33,63)(34,59)(35,58)(36,60)(37,74)(38,73)(39,75)(40,80)(41,79)(42,81)(43,77)(44,76)(45,78)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69);; 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*s1*s2*s1*s2,
s0*s1*s0*s1*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*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1,
s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)(28,55)(29,56)(30,57)(31,64)(32,65)(33,66)(34,73)(35,74)(36,75)(37,58)(38,59)(39,60)(40,67)(41,68)(42,69)(43,76)(44,77)(45,78)(46,61)(47,62)(48,63)(49,70)(50,71)(51,72)(52,79)(53,80)(54,81); s1 := 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); s2 := Sym(81)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(28,56)(29,55)(30,57)(31,62)(32,61)(33,63)(34,59)(35,58)(36,60)(37,74)(38,73)(39,75)(40,80)(41,79)(42,81)(43,77)(44,76)(45,78)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69); 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*s1*s2*s1*s2, s0*s1*s0*s1*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*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0 >;
References
None.
to this polytope.