Polytope of Type {42,4} with group SmallGroup(1344,6453)

Overview

Group: SmallGroup(1344,6453)
Rank: 3
Schläfli Type: {42,4}
Vertices, edges, …: 168, 336, 16
Order of s₀s₁s₂: 42
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Self-Petrie

Quotients maximal quotients in bold

4-fold

{42,4}*336b

7-fold

{6,4}*192a

8-fold

{21,4}*168

28-fold

{6,4}*48c

56-fold

{3,4}*24

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)^21> of order 2

10 facets

4 of {21}*42
6 of {42}*84

84 vertex figures

84 of {4}*8

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

8 facets

8 of {42}*84

84 vertex figures

84 of {4}*8

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

8 facets

8 of {42}*84

98 vertex figures

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

P/N, where N=<s0*s1*s2*s1*s0*(s1*s2)^2, (s0*s1)^21> of order 4

6 facets

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

42 vertex figures

42 of {4}*8

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

4 facets

4 of {42}*84

56 vertex figures

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

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

4 facets

4 of {42}*84

63 vertex figures

42 of {2}*4
21 of {4}*8

Representations

Permutation Representation (GAP)

s0 := (  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 97)( 18, 98)( 19,100)( 20, 99)( 21,101)( 22,102)( 23,104)( 24,103)( 25,109)( 26,110)( 27,112)( 28,111)( 29,105)( 30,106)( 31,108)( 32,107)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 93)( 42, 94)( 43, 96)( 44, 95)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 77)( 58, 78)( 59, 80)( 60, 79)( 61, 73)( 62, 74)( 63, 76)( 64, 75);;
s1 := (  1, 17)(  2, 20)(  3, 19)(  4, 18)(  5, 32)(  6, 29)(  7, 30)(  8, 31)(  9, 27)( 10, 26)( 11, 25)( 12, 28)( 13, 22)( 14, 23)( 15, 24)( 16, 21)( 33, 97)( 34,100)( 35, 99)( 36, 98)( 37,112)( 38,109)( 39,110)( 40,111)( 41,107)( 42,106)( 43,105)( 44,108)( 45,102)( 46,103)( 47,104)( 48,101)( 49, 81)( 50, 84)( 51, 83)( 52, 82)( 53, 96)( 54, 93)( 55, 94)( 56, 95)( 57, 91)( 58, 90)( 59, 89)( 60, 92)( 61, 86)( 62, 87)( 63, 88)( 64, 85)( 66, 68)( 69, 80)( 70, 77)( 71, 78)( 72, 79)( 73, 75);;
s2 := (  1,  5)(  2,  6)(  3,  7)(  4,  8)(  9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)(108,112);;
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*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(112)!(  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 97)( 18, 98)( 19,100)( 20, 99)( 21,101)( 22,102)( 23,104)( 24,103)( 25,109)( 26,110)( 27,112)( 28,111)( 29,105)( 30,106)( 31,108)( 32,107)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 93)( 42, 94)( 43, 96)( 44, 95)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 77)( 58, 78)( 59, 80)( 60, 79)( 61, 73)( 62, 74)( 63, 76)( 64, 75);
s1 := Sym(112)!(  1, 17)(  2, 20)(  3, 19)(  4, 18)(  5, 32)(  6, 29)(  7, 30)(  8, 31)(  9, 27)( 10, 26)( 11, 25)( 12, 28)( 13, 22)( 14, 23)( 15, 24)( 16, 21)( 33, 97)( 34,100)( 35, 99)( 36, 98)( 37,112)( 38,109)( 39,110)( 40,111)( 41,107)( 42,106)( 43,105)( 44,108)( 45,102)( 46,103)( 47,104)( 48,101)( 49, 81)( 50, 84)( 51, 83)( 52, 82)( 53, 96)( 54, 93)( 55, 94)( 56, 95)( 57, 91)( 58, 90)( 59, 89)( 60, 92)( 61, 86)( 62, 87)( 63, 88)( 64, 85)( 66, 68)( 69, 80)( 70, 77)( 71, 78)( 72, 79)( 73, 75);
s2 := Sym(112)!(  1,  5)(  2,  6)(  3,  7)(  4,  8)(  9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)(108,112);
poly := sub<Sym(112)|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*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.