Polytope of Type {21,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,4,4}*1344a
if this polytope has a name.
Group : SmallGroup(1344,6453)
Rank : 4
Schlafli Type : {21,4,4}
Number of vertices, edges, etc : 21, 84, 16, 8
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   4-fold quotients : {21,4,2}*336
   7-fold quotients : {3,4,4}*192a
   28-fold quotients : {3,4,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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