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

Overview

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

{2,14}*56

36-fold

{2,7}*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*(s2*s1)^2)^2*s0*(s2*s1)^3> of order 2

63 facets

63 of {4}*8

9 vertex figures

9 of {28}*56

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

63 facets

63 of {4}*8

9 vertex figures

9 of {28}*56

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

70 facets

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

10 vertex figures

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

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

42 facets

42 of {4}*8

6 vertex figures

6 of {28}*56

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

35 facets

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

5 vertex figures

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

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

28 facets

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

4 vertex figures

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

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.