Polytope of Type {28,4} with group SmallGroup(1008,896)

Overview

Group: SmallGroup(1008,896)
Rank: 3
Schläfli Type: {28,4}
Vertices, edges, …: 126, 252, 18
Order of s₀s₁s₂: 42
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

{4,4}*144

14-fold

{4,4}*72

18-fold

{14,2}*56

36-fold

{7,2}*28

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)^2)^2*s2*s1> of order 2

9 facets

9 of {28}*56

63 vertex figures

63 of {4}*8

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

9 facets

9 of {28}*56

63 vertex figures

63 of {4}*8

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

10 facets

2 of {14}*28
8 of {28}*56

70 vertex figures

56 of {4}*8
14 of {2}*4

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

6 facets

6 of {28}*56

42 vertex figures

42 of {4}*8

P/N, where N=<(s2*s1*s0)^2*s1*(s2*s1*s0)^2*s2*s1*s2, (s0*s1)^3*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 4

5 facets

1 of {14}*28
4 of {28}*56

35 vertex figures

28 of {4}*8
7 of {2}*4

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

4 facets

2 of {14}*28
2 of {28}*56

28 vertex figures

14 of {4}*8
14 of {2}*4

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.