Overview
- Group
- SmallGroup(216,101)
- Rank
- 3
- Schläfli Type
- {6,18}
- Vertices, edges, …
- 6, 54, 18
- Order of s0s1s2
- 18
- Order of s0s1s2s1
- 6
- Also known as
- if this polytope has a name.
Special Properties
- Compact Hyperbolic Quotient
- Locally Spherical
- Orientable
- Flat
Quotients maximal quotients in bold
2-fold
3-fold
6-fold
9-fold
18-fold
27-fold
Covers minimal covers in bold
2-fold
3-fold
4-fold
5-fold
6-fold
- {18,36}*1296b
- {6,36}*1296a
- {6,108}*1296b
- {36,18}*1296c
- {12,18}*1296e
- {12,54}*1296b
- {6,36}*1296l
- {12,18}*1296l
7-fold
8-fold
- {6,144}*1728b
- {24,36}*1728a
- {12,36}*1728b
- {24,36}*1728b
- {12,72}*1728b
- {12,72}*1728d
- {48,18}*1728b
- {6,36}*1728a
- {12,18}*1728a
- {6,18}*1728a
- {6,36}*1728c
- {12,18}*1728b
- {12,36}*1728f
- {12,36}*1728g
- {24,18}*1728b
- {24,18}*1728d
- {12,18}*1728d
9-fold
Irregular Quotients of which this is a minimal cover
None.
Representations
Permutation Representation (GAP)
s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);; s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,23)(11,22)(12,24)(13,20)(14,19)(15,21)(16,26)(17,25)(18,27)(28,31)(29,33)(30,32)(35,36)(37,50)(38,49)(39,51)(40,47)(41,46)(42,48)(43,53)(44,52)(45,54);; s2 := ( 1,37)( 2,39)( 3,38)( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,47)(20,46)(21,48)(22,53)(23,52)(24,54)(25,50)(26,49)(27,51);; 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*s2*s0*s1*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(54)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54); s1 := Sym(54)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,23)(11,22)(12,24)(13,20)(14,19)(15,21)(16,26)(17,25)(18,27)(28,31)(29,33)(30,32)(35,36)(37,50)(38,49)(39,51)(40,47)(41,46)(42,48)(43,53)(44,52)(45,54); s2 := Sym(54)!( 1,37)( 2,39)( 3,38)( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,47)(20,46)(21,48)(22,53)(23,52)(24,54)(25,50)(26,49)(27,51); poly := sub<Sym(54)|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*s2*s0*s1*s0*s1*s2*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References
None.
to this polytope.