Overview
- Group
- SmallGroup(648,555)
- Rank
- 4
- Schläfli Type
- {3,6,6}
- Vertices, edges, …
- 3, 27, 54, 18
- Order of s0s1s2s3
- 6
- Order of s0s1s2s3s2s1
- 6
- Also known as
- if this polytope has a name.
Special Properties
- Universal
- Orientable
- Flat
Quotients maximal quotients in bold
3-fold
6-fold
9-fold
18-fold
27-fold
Covers minimal covers in bold
2-fold
3-fold
Irregular Quotients of which this is a minimal cover
Click an entry to reveal its facets and vertex figures.
P/N, where N=<(s2*s3)^3> of order 2
9 facets
- 9 of {3,6}*36
3 vertex figures
- 3 of 2-fold non-regular quotient of {6,6}*216d
P/N, where N=<(s2*s3)^2> of order 3
6 facets
- 6 of {3,6}*36
3 vertex figures
- 2 of 3-fold non-regular quotient of {6,6}*216d
- 1 of {6,6}*72c
P/N, where N=<(s1*s2)^2> of order 3
12 facets
3 vertex figures
- 3 of 3-fold non-regular quotient of {6,6}*216d
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);; s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,15)(11,14)(12,13)(16,18)(19,23)(20,22)(21,24)(25,26);; s2 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)(22,25)(23,27)(24,26);; s3 := ( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26);; poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP)
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26); s1 := Sym(27)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,15)(11,14)(12,13)(16,18)(19,23)(20,22)(21,24)(25,26); s2 := Sym(27)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)(22,25)(23,27)(24,26); s3 := Sym(27)!( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26); poly := sub<Sym(27)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma)
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s1*s2 >;
References
None.
to this polytope.