Overview
- Group
- SmallGroup(1296,3529)
- Rank
- 3
- Schläfli Type
- {6,12}
- Vertices, edges, …
- 54, 324, 108
- Order of s0s1s2
- 12
- 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
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=<s1*(s0*(s2*s1)^2)^2*s0*s2*s1*s2> of order 2
54 facets
- 54 of {6}*12
27 vertex figures
- 27 of {12}*24
P/N, where N=<s0*(s2*s1)^2*s0*s2*s1*s0*(s2*s1)^2*s2> of order 2
57 facets
27 vertex figures
- 27 of {12}*24
P/N, where N=<s0*s1*(s0*(s2*s1)^2)^2*s0*s2*s1*s2> of order 2
54 facets
- 54 of {6}*12
30 vertex figures
P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2> of order 3
36 facets
- 36 of {6}*12
18 vertex figures
- 18 of {12}*24
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2> of order 3
36 facets
- 36 of {6}*12
18 vertex figures
- 18 of {12}*24
P/N, where N=<(s0*(s1*s2)^2*s1)^2> of order 3
36 facets
- 36 of {6}*12
18 vertex figures
- 18 of {12}*24
P/N, where N=<s0*s2*(s1*s0)^2*(s1*s2)^2*s1*s0*s2*s1> of order 3
36 facets
- 36 of {6}*12
18 vertex figures
- 18 of {12}*24
P/N, where N=<(s0*s1)^3, s0*(s2*s1)^2*s0*s2*s1*s0*(s2*s1)^2*s2> of order 4
30 facets
15 vertex figures
P/N, where N=<(s0*(s1*s2)^2*s1)^2, s1*s0*(s2*s1)^3*s0*(s2*s1)^2*s2> of order 6
18 facets
- 18 of {6}*12
12 vertex figures
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, (s1*s2)^6> of order 6
18 facets
- 18 of {6}*12
12 vertex figures
P/N, where N=<s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2, s0*(s2*s1)^2*s0*s2*s1*s0*(s2*s1)^2*s2> of order 6
21 facets
9 vertex figures
- 9 of {12}*24
P/N, where N=<(s0*s1)^2, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 9
18 facets
6 vertex figures
- 6 of {12}*24
P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s0*(s1*s2)^3*s1*s0*s2> of order 9
12 facets
- 12 of {6}*12
12 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,73)(38,74)(39,75)(40,79)(41,80)(42,81)(43,76)(44,77)(45,78)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69);; s1 := ( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,40)(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,49)(20,51)(21,50)(22,46)(23,48)(24,47)(25,52)(26,54)(27,53)(55,58)(56,60)(57,59)(62,63)(64,67)(65,69)(66,68)(71,72)(73,76)(74,78)(75,77)(80,81);; s2 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,29)(11,28)(12,30)(13,35)(14,34)(15,36)(16,32)(17,31)(18,33)(19,56)(20,55)(21,57)(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(37,38)(40,44)(41,43)(42,45)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69)(73,74)(76,80)(77,79)(78,81);; 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,
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,73)(38,74)(39,75)(40,79)(41,80)(42,81)(43,76)(44,77)(45,78)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69); s1 := Sym(81)!( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,40)(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,49)(20,51)(21,50)(22,46)(23,48)(24,47)(25,52)(26,54)(27,53)(55,58)(56,60)(57,59)(62,63)(64,67)(65,69)(66,68)(71,72)(73,76)(74,78)(75,77)(80,81); s2 := Sym(81)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,29)(11,28)(12,30)(13,35)(14,34)(15,36)(16,32)(17,31)(18,33)(19,56)(20,55)(21,57)(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(37,38)(40,44)(41,43)(42,45)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69)(73,74)(76,80)(77,79)(78,81); 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, s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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 >;
References
None.
to this polytope.