Polytope of Type {42,4} with group SmallGroup(1008,896)

Overview

Group: SmallGroup(1008,896)
Rank: 3
Schläfli Type: {42,4}
Vertices, edges, …: 126, 252, 12
Order of s₀s₁s₂: 28
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

7-fold

{6,4}*144

9-fold

{14,4}*112

14-fold

{6,4}*72

18-fold

{14,2}*56

36-fold

{7,2}*28

63-fold

{2,4}*16

126-fold

{2,2}*8

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*s0*s2*(s1*s0)^3*s2*s1*s0*s1> of order 2

7 facets

5 of {42}*84
2 of {21}*42

63 vertex figures

63 of {4}*8

P/N, where N=<(s0*s1)^3*s2*(s1*s0)^2*s1*s2> of order 2

6 facets

6 of {42}*84

63 vertex figures

63 of {4}*8

P/N, where N=<s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s2*s1*s2> of order 3

8 facets

2 of {42}*84
6 of {14}*28

42 vertex figures

42 of {4}*8

P/N, where N=<(s0*s1*s2*s1)^2> of order 3

4 facets

4 of {42}*84

42 vertex figures

42 of {4}*8

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, (s0*s1)^4*s0*s2*(s1*s0)^2*s1*s2> of order 6

5 facets

3 of {14}*28
2 of {21}*42

21 vertex figures

21 of {4}*8

Representations

Permutation Representation (GAP)

s0 := ( 2, 7)( 3, 6)( 4, 5)( 8,15)( 9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,57)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,50)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51);;
s1 := ( 1,23)( 2,22)( 3,28)( 4,27)( 5,26)( 6,25)( 7,24)( 8,30)( 9,29)(10,35)(11,34)(12,33)(13,32)(14,31)(15,37)(16,36)(17,42)(18,41)(19,40)(20,39)(21,38)(43,44)(45,49)(46,48)(50,51)(52,56)(53,55)(57,58)(59,63)(60,62);;
s2 := ( 8,22)( 9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56);;
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, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(63)!( 2, 7)( 3, 6)( 4, 5)( 8,15)( 9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,57)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,50)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51);
s1 := Sym(63)!( 1,23)( 2,22)( 3,28)( 4,27)( 5,26)( 6,25)( 7,24)( 8,30)( 9,29)(10,35)(11,34)(12,33)(13,32)(14,31)(15,37)(16,36)(17,42)(18,41)(19,40)(20,39)(21,38)(43,44)(45,49)(46,48)(50,51)(52,56)(53,55)(57,58)(59,63)(60,62);
s2 := Sym(63)!( 8,22)( 9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56);
poly := sub<Sym(63)|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, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0 >;

References

None.

to this polytope.