Polytope of Type {4,6,6} with group SmallGroup(864,4686)

Overview

Group: SmallGroup(864,4686)
Rank: 4
Schläfli Type: {4,6,6}
Vertices, edges, …: 12, 36, 54, 6
Order of s₀s₁s₂s₃: 12
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{4,6}₄,{6,6|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

{4,6,2}*288

6-fold

{4,6,2}*144

9-fold

{4,2,6}*96

18-fold

27-fold

{4,2,2}*32

36-fold

{2,2,3}*24

54-fold

{2,2,2}*16

Covers minimal covers in bold

2-fold

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*s1*s2)^2> of order 3

6 facets

6 of 3-fold non-regular quotient of {4,6}*144

4 vertex figures

4 of {6,6}*72a

P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3

6 facets

6 of 3-fold non-regular quotient of {4,6}*144

8 vertex figures

2 of {6,6}*72a
6 of {2,6}*24

Representations

Permutation Representation (GAP)

s0 := ( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)(31,37)(32,38)(33,39)(34,46)(35,47)(36,48)(43,49)(44,50)(45,51);;
s1 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);;
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)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,43)(35,45)(36,44)(46,49)(47,51)(48,50)(53,54);;
s3 := ( 1,29)( 2,28)( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,38)(11,37)(12,39)(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,47)(20,46)(21,48)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54);;
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*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(54)!( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)(31,37)(32,38)(33,39)(34,46)(35,47)(36,48)(43,49)(44,50)(45,51);
s1 := Sym(54)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);
s2 := Sym(54)!( 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)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,43)(35,45)(36,44)(46,49)(47,51)(48,50)(53,54);
s3 := Sym(54)!( 1,29)( 2,28)( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,38)(11,37)(12,39)(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,47)(20,46)(21,48)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54);
poly := sub<Sym(54)|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*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.