Overview
- Group
- SmallGroup(648,555)
- Rank
- 4
- Schläfli Type
- {6,6,3}
- Vertices, edges, …
- 18, 54, 27, 3
- 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=<(s0*s1)^3> of order 2
3 facets
- 3 of 2-fold non-regular quotient of {6,6}*216d
9 vertex figures
- 9 of {6,3}*36
P/N, where N=<(s0*s1)^2> of order 3
3 facets
- 2 of 3-fold non-regular quotient of {6,6}*216d
- 1 of {6,6}*72b
6 vertex figures
- 6 of {6,3}*36
P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3
3 facets
- 3 of 3-fold non-regular quotient of {6,6}*216d
12 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27);; s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,12)(13,18)(14,17)(15,16)(19,20)(22,26)(23,25)(24,27);; s2 := ( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)(20,24)(21,23)(26,27);; s3 := ( 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);; 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, s2*s3*s2*s3*s2*s3,
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(27)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27); s1 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,12)(13,18)(14,17)(15,16)(19,20)(22,26)(23,25)(24,27); s2 := Sym(27)!( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)(20,24)(21,23)(26,27); s3 := Sym(27)!( 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); 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, s2*s3*s2*s3*s2*s3, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s0*s1 >;
References
None.
to this polytope.