Polytope of Type {21,4,4} with group SmallGroup(1344,6453)

Overview

Group: SmallGroup(1344,6453)
Rank: 4
Schläfli Type: {21,4,4}
Vertices, edges, …: 21, 84, 16, 8
Order of s₀s₁s₂s₃: 42
Order of s₀s₁s₂s₃s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Universal
Non-Orientable
Flat

Quotients maximal quotients in bold

4-fold

{21,4,2}*336

7-fold

{3,4,4}*192a

28-fold

{3,4,2}*48

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=<(s2*s3)^2> of order 2

4 facets

4 of {21,4}*168

21 vertex figures

14 of 2-fold non-regular quotient of {4,4}*64
7 of {4,4}*32

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, 29)(  6, 32)(  7, 31)(  8, 30)(  9, 25)( 10, 28)( 11, 27)( 12, 26)( 13, 21)( 14, 24)( 15, 23)( 16, 22)( 33, 97)( 34,100)( 35, 99)( 36, 98)( 37,109)( 38,112)( 39,111)( 40,110)( 41,105)( 42,108)( 43,107)( 44,106)( 45,101)( 46,104)( 47,103)( 48,102)( 49, 81)( 50, 84)( 51, 83)( 52, 82)( 53, 93)( 54, 96)( 55, 95)( 56, 94)( 57, 89)( 58, 92)( 59, 91)( 60, 90)( 61, 85)( 62, 88)( 63, 87)( 64, 86)( 66, 68)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 74, 76);;
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);;
s3 := (  5,  6)(  7,  8)(  9, 11)( 10, 12)( 13, 16)( 14, 15)( 21, 22)( 23, 24)( 25, 27)( 26, 28)( 29, 32)( 30, 31)( 37, 38)( 39, 40)( 41, 43)( 42, 44)( 45, 48)( 46, 47)( 53, 54)( 55, 56)( 57, 59)( 58, 60)( 61, 64)( 62, 63)( 69, 70)( 71, 72)( 73, 75)( 74, 76)( 77, 80)( 78, 79)( 85, 86)( 87, 88)( 89, 91)( 90, 92)( 93, 96)( 94, 95)(101,102)(103,104)(105,107)(106,108)(109,112)(110,111);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s1*s0*s2*s1*s2*s1*s0*s1, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*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 ];;
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, 29)(  6, 32)(  7, 31)(  8, 30)(  9, 25)( 10, 28)( 11, 27)( 12, 26)( 13, 21)( 14, 24)( 15, 23)( 16, 22)( 33, 97)( 34,100)( 35, 99)( 36, 98)( 37,109)( 38,112)( 39,111)( 40,110)( 41,105)( 42,108)( 43,107)( 44,106)( 45,101)( 46,104)( 47,103)( 48,102)( 49, 81)( 50, 84)( 51, 83)( 52, 82)( 53, 93)( 54, 96)( 55, 95)( 56, 94)( 57, 89)( 58, 92)( 59, 91)( 60, 90)( 61, 85)( 62, 88)( 63, 87)( 64, 86)( 66, 68)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 74, 76);
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);
s3 := Sym(112)!(  5,  6)(  7,  8)(  9, 11)( 10, 12)( 13, 16)( 14, 15)( 21, 22)( 23, 24)( 25, 27)( 26, 28)( 29, 32)( 30, 31)( 37, 38)( 39, 40)( 41, 43)( 42, 44)( 45, 48)( 46, 47)( 53, 54)( 55, 56)( 57, 59)( 58, 60)( 61, 64)( 62, 63)( 69, 70)( 71, 72)( 73, 75)( 74, 76)( 77, 80)( 78, 79)( 85, 86)( 87, 88)( 89, 91)( 90, 92)( 93, 96)( 94, 95)(101,102)(103,104)(105,107)(106,108)(109,112)(110,111);
poly := sub<Sym(112)|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, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s1*s0*s2*s1*s2*s1*s0*s1, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*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 >;

References

None.

to this polytope.