Polytope of Type {4,21,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,21,4}*672
if this polytope has a name.
Group : SmallGroup(672,1265)
Rank : 4
Schlafli Type : {4,21,4}
Number of vertices, edges, etc : 4, 42, 42, 4
Order of s₀s₁s₂s₃ : 21
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,21,4,2} of size 1344
Vertex Figure Of :
   {2,4,21,4} of size 1344
Quotients (Maximal Quotients in Boldface) :
   7-fold quotients : {4,3,4}*96
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,21,4}*1344a, {4,21,4}*1344b, {4,42,4}*1344d, {4,42,4}*1344e, {4,42,4}*1344f, {4,42,4}*1344g
Permutation Representation (GAP) :

s0 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 97)( 18, 99)
( 19, 98)( 20,100)( 21,105)( 22,107)( 23,106)( 24,108)( 25,101)( 26,103)
( 27,102)( 28,104)( 29,109)( 30,111)( 31,110)( 32,112)( 33, 81)( 34, 83)
( 35, 82)( 36, 84)( 37, 89)( 38, 91)( 39, 90)( 40, 92)( 41, 85)( 42, 87)
( 43, 86)( 44, 88)( 45, 93)( 46, 95)( 47, 94)( 48, 96)( 49, 65)( 50, 67)
( 51, 66)( 52, 68)( 53, 73)( 54, 75)( 55, 74)( 56, 76)( 57, 69)( 58, 71)
( 59, 70)( 60, 72)( 61, 77)( 62, 79)( 63, 78)( 64, 80);;
s2 := (  1, 17)(  2, 18)(  3, 20)(  4, 19)(  5, 29)(  6, 30)(  7, 32)(  8, 31)
(  9, 25)( 10, 26)( 11, 28)( 12, 27)( 13, 21)( 14, 22)( 15, 24)( 16, 23)
( 33, 97)( 34, 98)( 35,100)( 36, 99)( 37,109)( 38,110)( 39,112)( 40,111)
( 41,105)( 42,106)( 43,108)( 44,107)( 45,101)( 46,102)( 47,104)( 48,103)
( 49, 81)( 50, 82)( 51, 84)( 52, 83)( 53, 93)( 54, 94)( 55, 96)( 56, 95)
( 57, 89)( 58, 90)( 59, 92)( 60, 91)( 61, 85)( 62, 86)( 63, 88)( 64, 87)
( 67, 68)( 69, 77)( 70, 78)( 71, 80)( 72, 79)( 75, 76);;
s3 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(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, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s3*s2*s1*s3*s2*s3*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(112)!(  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)
(  8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112);
s1 := Sym(112)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 97)
( 18, 99)( 19, 98)( 20,100)( 21,105)( 22,107)( 23,106)( 24,108)( 25,101)
( 26,103)( 27,102)( 28,104)( 29,109)( 30,111)( 31,110)( 32,112)( 33, 81)
( 34, 83)( 35, 82)( 36, 84)( 37, 89)( 38, 91)( 39, 90)( 40, 92)( 41, 85)
( 42, 87)( 43, 86)( 44, 88)( 45, 93)( 46, 95)( 47, 94)( 48, 96)( 49, 65)
( 50, 67)( 51, 66)( 52, 68)( 53, 73)( 54, 75)( 55, 74)( 56, 76)( 57, 69)
( 58, 71)( 59, 70)( 60, 72)( 61, 77)( 62, 79)( 63, 78)( 64, 80);
s2 := Sym(112)!(  1, 17)(  2, 18)(  3, 20)(  4, 19)(  5, 29)(  6, 30)(  7, 32)
(  8, 31)(  9, 25)( 10, 26)( 11, 28)( 12, 27)( 13, 21)( 14, 22)( 15, 24)
( 16, 23)( 33, 97)( 34, 98)( 35,100)( 36, 99)( 37,109)( 38,110)( 39,112)
( 40,111)( 41,105)( 42,106)( 43,108)( 44,107)( 45,101)( 46,102)( 47,104)
( 48,103)( 49, 81)( 50, 82)( 51, 84)( 52, 83)( 53, 93)( 54, 94)( 55, 96)
( 56, 95)( 57, 89)( 58, 90)( 59, 92)( 60, 91)( 61, 85)( 62, 86)( 63, 88)
( 64, 87)( 67, 68)( 69, 77)( 70, 78)( 71, 80)( 72, 79)( 75, 76);
s3 := Sym(112)!(  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(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, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s3*s2*s1*s3*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope