Overview
- Group
- SmallGroup(1728,47847)
- Rank
- 3
- Schläfli Type
- {12,4}
- Vertices, edges, …
- 216, 432, 72
- Order of s0s1s2
- 6
- Order of s0s1s2s1
- 12
- Also known as
- if this polytope has a name.
Special Properties
- Compact Hyperbolic Quotient
- Locally Spherical
- Orientable
Quotients maximal quotients in bold
4-fold
9-fold
12-fold
18-fold
24-fold
36-fold
72-fold
144-fold
216-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)^3*s1*s2*s1*s0*s1> of order 2
36 facets
- 36 of {12}*24
108 vertex figures
- 108 of {4}*8
P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2*(s1*s0)^2*s2*s1> of order 2
36 facets
- 36 of {12}*24
108 vertex figures
- 108 of {4}*8
P/N, where N=<s0*s1*(s2*(s1*s0)^2)^2*s2> of order 2
36 facets
- 36 of {12}*24
108 vertex figures
- 108 of {4}*8
P/N, where N=<s0*s1*s0*(s2*(s1*s0)^2)^2*s2*s1> of order 2
36 facets
- 36 of {12}*24
108 vertex figures
- 108 of {4}*8
P/N, where N=<((s1*s0)^2*s2)^2*s1*s0*s1*s2> of order 2
36 facets
- 36 of {12}*24
108 vertex figures
- 108 of {4}*8
P/N, where N=<s0*s1*s2*(s1*s0)^5*s1*s2*s1> of order 3
24 facets
- 24 of {12}*24
72 vertex figures
- 72 of {4}*8
P/N, where N=<(s0*s1)^2*s0*s2*(s1*s0)^3*s2*s1> of order 3
24 facets
- 24 of {12}*24
72 vertex figures
- 72 of {4}*8
P/N, where N=<(s0*s1)^6, (s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 4
20 facets
54 vertex figures
- 54 of {4}*8
P/N, where N=<(s0*s1)^6, s0*s1*s0*s2*(s1*s0)^4*s2*s1> of order 4
20 facets
56 vertex figures
P/N, where N=<(s1*s0)^2*s2*(s1*s0)^3*s1*s2> of order 4
18 facets
- 18 of {12}*24
54 vertex figures
- 54 of {4}*8
P/N, where N=<(s0*s1)^2*s0*s2*(s1*s0)^3*s1*s2, s0*s2*(s1*s0)^4*s2*s1*s0*s1> of order 4
18 facets
- 18 of {12}*24
58 vertex figures
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*(s1*s0)^2*s2*s1, ((s1*s0)^2*s2)^2*s1*s0*s1*s2> of order 4
18 facets
- 18 of {12}*24
54 vertex figures
- 54 of {4}*8
P/N, where N=<s0*s2*(s1*s0)^4*s2*s1*s0*s1, (s0*s1)^3*(s2*s1*s0*s1)^2*s2> of order 4
18 facets
- 18 of {12}*24
56 vertex figures
P/N, where N=<s0*s1*(s2*(s1*s0)^2)^2*s2, (s0*s1)^2*s2*s1*s0*s1*s2*(s1*s0)^2*s2*s1> of order 4
18 facets
- 18 of {12}*24
54 vertex figures
- 54 of {4}*8
P/N, where N=<(s0*s1)^6, (s0*s1)^2*s2*s1*s0*s1*s2*(s1*s0)^2*s2*s1> of order 4
20 facets
54 vertex figures
- 54 of {4}*8
P/N, where N=<s0*s1*(s2*(s1*s0)^2)^2*s2, s0*s1*s0*s2*(s1*s0)^4*s2*s1> of order 4
18 facets
- 18 of {12}*24
56 vertex figures
P/N, where N=<s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2, (s0*s1)^6> of order 4
20 facets
54 vertex figures
- 54 of {4}*8
P/N, where N=<(s0*s1)^2*s0*(s2*s1*s0*s1)^2*s2*s1, s0*(s2*s1*s0*s1)^2*s2*(s1*s0)^2*s1> of order 4
18 facets
- 18 of {12}*24
54 vertex figures
- 54 of {4}*8
P/N, where N=<s0*s1*(s2*(s1*s0)^2)^2*s2, (s0*s1)^2*s0*s2*(s1*s0)^3*s2*s1> of order 6
12 facets
- 12 of {12}*24
36 vertex figures
- 36 of {4}*8
P/N, where N=<(s0*s1)^4, s0*s1*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 6
18 facets
36 vertex figures
- 36 of {4}*8
P/N, where N=<(s0*s1)^4, (s0*s1)^2*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 6
18 facets
36 vertex figures
- 36 of {4}*8
P/N, where N=<s0*(s1*s0*s2)^2*s1*s2, (s1*s0)^2*s2*s1*s0*(s1*s2)^2> of order 6
12 facets
- 12 of {12}*24
36 vertex figures
- 36 of {4}*8
P/N, where N=<(s0*s1)^3*s0*s2*(s1*s0)^2*s1*s2, (s0*s1)^2*s2*s1*s0*s1*s2*(s1*s0)^2*s2*s1> of order 6
12 facets
- 12 of {12}*24
36 vertex figures
- 36 of {4}*8
P/N, where N=<s0*s1*s0*(s2*(s1*s0)^2)^2*s2*s1, (s1*s0*s1*s2)^4> of order 6
12 facets
- 12 of {12}*24
36 vertex figures
- 36 of {4}*8
P/N, where N=<(s0*s1)^6, s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s1*s2> of order 6
16 facets
36 vertex figures
- 36 of {4}*8
P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s1*(s2*s1*s0)^2*s2> of order 6
12 facets
- 12 of {12}*24
40 vertex figures
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2, (s0*s1)^2*s0*s2*(s1*s0)^3*s2*s1> of order 6
12 facets
- 12 of {12}*24
40 vertex figures
P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 6
12 facets
- 12 of {12}*24
36 vertex figures
- 36 of {4}*8
P/N, where N=<s0*s1*s0*(s2*(s1*s0)^2)^2*s2*s1, s0*(s1*s0*s2)^2*(s1*s0)^3*s1*s2> of order 6
12 facets
- 12 of {12}*24
36 vertex figures
- 36 of {4}*8
P/N, where N=<(s0*s1)^6, s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s1> of order 6
16 facets
36 vertex figures
- 36 of {4}*8
P/N, where N=<s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2, (s0*s1)^6, s0*s1*s0*s2*(s1*s0)^4*s2*s1> of order 8
10 facets
28 vertex figures
P/N, where N=<(s0*s1)^6, s0*s1*s0*s2*(s1*s0)^4*s2*s1, (s0*s1)^3*(s2*s1*s0*s1)^2*s2> of order 8
10 facets
28 vertex figures
P/N, where N=<(s0*s1)^3, s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 8
12 facets
27 vertex figures
- 27 of {4}*8
P/N, where N=<(s0*s1)^6, (s1*s0)^2*s2*(s1*s0)^3*s1*s2> of order 8
10 facets
28 vertex figures
P/N, where N=<s0*s1*(s2*(s1*s0)^2)^2*s2, (s0*s1)^2*s0*s2*(s1*s0)^3*s1*s2, s0*s2*(s1*s0)^4*s2*s1*s0*s1> of order 8
9 facets
- 9 of {12}*24
29 vertex figures
P/N, where N=<(s0*s1)^4, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9
12 facets
24 vertex figures
- 24 of {4}*8
P/N, where N=<(s0*s2*s1)^3, s1*s2*(s1*s0)^2*s2*s1*s0*s1*s2, (s0*s1)^6> of order 12
8 facets
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2, s0*s1*(s2*(s1*s0)^2)^2*s2, (s0*s1)^2*s0*s2*(s1*s0)^3*s2*s1> of order 12
6 facets
- 6 of {12}*24
20 vertex figures
P/N, where N=<(s0*s1*s2*s1)^2*s0*(s1*s2)^2, (s1*s0)^4*s2*s1*s0*s1*s2> of order 12
6 facets
- 6 of {12}*24
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s0*s1)^6, s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s1*s2, (s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 12
8 facets
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s0*s1*s2*s1)^2, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 12
8 facets
20 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 3, 4)( 5,29)( 6,30)( 7,32)( 8,31)( 9,21)(10,22)(11,24)(12,23)(13,25)(14,26)(15,28)(16,27)(19,20)(35,36);; s1 := ( 2, 4)( 6, 8)(10,12)(13,33)(14,36)(15,35)(16,34)(17,25)(18,28)(19,27)(20,26)(21,29)(22,32)(23,31)(24,30);; s2 := ( 1,18)( 2,17)( 3,20)( 4,19)( 5,26)( 6,25)( 7,28)( 8,27)( 9,10)(11,12)(13,30)(14,29)(15,32)(16,31)(21,22)(23,24)(33,34)(35,36);; 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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(36)!( 3, 4)( 5,29)( 6,30)( 7,32)( 8,31)( 9,21)(10,22)(11,24)(12,23)(13,25)(14,26)(15,28)(16,27)(19,20)(35,36); s1 := Sym(36)!( 2, 4)( 6, 8)(10,12)(13,33)(14,36)(15,35)(16,34)(17,25)(18,28)(19,27)(20,26)(21,29)(22,32)(23,31)(24,30); s2 := Sym(36)!( 1,18)( 2,17)( 3,20)( 4,19)( 5,26)( 6,25)( 7,28)( 8,27)( 9,10)(11,12)(13,30)(14,29)(15,32)(16,31)(21,22)(23,24)(33,34)(35,36); poly := sub<Sym(36)|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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1 >;
References
None.
to this polytope.