Polytope of Type {2,2,6,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,6,6,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,46165)
Rank : 6
Schlafli Type : {2,2,6,6,6}
Number of vertices, edges, etc : 2, 2, 6, 18, 18, 6
Order of s0s1s2s3s4s5 : 6
Order of s0s1s2s3s4s5s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
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) :
2-fold quotients : {2,2,3,6,6}*864a, {2,2,6,6,3}*864a
3-fold quotients : {2,2,6,2,6}*576
4-fold quotients : {2,2,3,6,3}*432
6-fold quotients : {2,2,3,2,6}*288, {2,2,6,2,3}*288
9-fold quotients : {2,2,2,2,6}*192, {2,2,6,2,2}*192
12-fold quotients : {2,2,3,2,3}*144
18-fold quotients : {2,2,2,2,3}*96, {2,2,3,2,2}*96
27-fold quotients : {2,2,2,2,2}*64
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 6, 7)( 8, 11)( 9, 13)( 10, 12)( 15, 16)( 17, 20)( 18, 22)( 19, 21)
( 24, 25)( 26, 29)( 27, 31)( 28, 30)( 33, 34)( 35, 38)( 36, 40)( 37, 39)
( 42, 43)( 44, 47)( 45, 49)( 46, 48)( 51, 52)( 53, 56)( 54, 58)( 55, 57)
( 60, 61)( 62, 65)( 63, 67)( 64, 66)( 69, 70)( 71, 74)( 72, 76)( 73, 75)
( 78, 79)( 80, 83)( 81, 85)( 82, 84)( 87, 88)( 89, 92)( 90, 94)( 91, 93)
( 96, 97)( 98,101)( 99,103)(100,102)(105,106)(107,110)(108,112)(109,111);;
s3 := ( 5, 62)( 6, 64)( 7, 63)( 8, 59)( 9, 61)( 10, 60)( 11, 65)( 12, 67)
( 13, 66)( 14, 71)( 15, 73)( 16, 72)( 17, 68)( 18, 70)( 19, 69)( 20, 74)
( 21, 76)( 22, 75)( 23, 80)( 24, 82)( 25, 81)( 26, 77)( 27, 79)( 28, 78)
( 29, 83)( 30, 85)( 31, 84)( 32, 89)( 33, 91)( 34, 90)( 35, 86)( 36, 88)
( 37, 87)( 38, 92)( 39, 94)( 40, 93)( 41, 98)( 42,100)( 43, 99)( 44, 95)
( 45, 97)( 46, 96)( 47,101)( 48,103)( 49,102)( 50,107)( 51,109)( 52,108)
( 53,104)( 54,106)( 55,105)( 56,110)( 57,112)( 58,111);;
s4 := ( 5, 14)( 6, 16)( 7, 15)( 8, 18)( 9, 17)( 10, 19)( 11, 22)( 12, 21)
( 13, 20)( 24, 25)( 26, 27)( 29, 31)( 32, 41)( 33, 43)( 34, 42)( 35, 45)
( 36, 44)( 37, 46)( 38, 49)( 39, 48)( 40, 47)( 51, 52)( 53, 54)( 56, 58)
( 59, 68)( 60, 70)( 61, 69)( 62, 72)( 63, 71)( 64, 73)( 65, 76)( 66, 75)
( 67, 74)( 78, 79)( 80, 81)( 83, 85)( 86, 95)( 87, 97)( 88, 96)( 89, 99)
( 90, 98)( 91,100)( 92,103)( 93,102)( 94,101)(105,106)(107,108)(110,112);;
s5 := ( 5, 32)( 6, 34)( 7, 33)( 8, 35)( 9, 37)( 10, 36)( 11, 38)( 12, 40)
( 13, 39)( 14, 50)( 15, 52)( 16, 51)( 17, 53)( 18, 55)( 19, 54)( 20, 56)
( 21, 58)( 22, 57)( 23, 41)( 24, 43)( 25, 42)( 26, 44)( 27, 46)( 28, 45)
( 29, 47)( 30, 49)( 31, 48)( 59, 86)( 60, 88)( 61, 87)( 62, 89)( 63, 91)
( 64, 90)( 65, 92)( 66, 94)( 67, 93)( 68,104)( 69,106)( 70,105)( 71,107)
( 72,109)( 73,108)( 74,110)( 75,112)( 76,111)( 77, 95)( 78, 97)( 79, 96)
( 80, 98)( 81,100)( 82, 99)( 83,101)( 84,103)( 85,102);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;; s5 := F.6;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5,
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s5*s0*s5,
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s5*s3*s4*s3*s4*s5*s3*s4*s3*s4,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3,
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(112)!(1,2);
s1 := Sym(112)!(3,4);
s2 := Sym(112)!( 6, 7)( 8, 11)( 9, 13)( 10, 12)( 15, 16)( 17, 20)( 18, 22)
( 19, 21)( 24, 25)( 26, 29)( 27, 31)( 28, 30)( 33, 34)( 35, 38)( 36, 40)
( 37, 39)( 42, 43)( 44, 47)( 45, 49)( 46, 48)( 51, 52)( 53, 56)( 54, 58)
( 55, 57)( 60, 61)( 62, 65)( 63, 67)( 64, 66)( 69, 70)( 71, 74)( 72, 76)
( 73, 75)( 78, 79)( 80, 83)( 81, 85)( 82, 84)( 87, 88)( 89, 92)( 90, 94)
( 91, 93)( 96, 97)( 98,101)( 99,103)(100,102)(105,106)(107,110)(108,112)
(109,111);
s3 := Sym(112)!( 5, 62)( 6, 64)( 7, 63)( 8, 59)( 9, 61)( 10, 60)( 11, 65)
( 12, 67)( 13, 66)( 14, 71)( 15, 73)( 16, 72)( 17, 68)( 18, 70)( 19, 69)
( 20, 74)( 21, 76)( 22, 75)( 23, 80)( 24, 82)( 25, 81)( 26, 77)( 27, 79)
( 28, 78)( 29, 83)( 30, 85)( 31, 84)( 32, 89)( 33, 91)( 34, 90)( 35, 86)
( 36, 88)( 37, 87)( 38, 92)( 39, 94)( 40, 93)( 41, 98)( 42,100)( 43, 99)
( 44, 95)( 45, 97)( 46, 96)( 47,101)( 48,103)( 49,102)( 50,107)( 51,109)
( 52,108)( 53,104)( 54,106)( 55,105)( 56,110)( 57,112)( 58,111);
s4 := Sym(112)!( 5, 14)( 6, 16)( 7, 15)( 8, 18)( 9, 17)( 10, 19)( 11, 22)
( 12, 21)( 13, 20)( 24, 25)( 26, 27)( 29, 31)( 32, 41)( 33, 43)( 34, 42)
( 35, 45)( 36, 44)( 37, 46)( 38, 49)( 39, 48)( 40, 47)( 51, 52)( 53, 54)
( 56, 58)( 59, 68)( 60, 70)( 61, 69)( 62, 72)( 63, 71)( 64, 73)( 65, 76)
( 66, 75)( 67, 74)( 78, 79)( 80, 81)( 83, 85)( 86, 95)( 87, 97)( 88, 96)
( 89, 99)( 90, 98)( 91,100)( 92,103)( 93,102)( 94,101)(105,106)(107,108)
(110,112);
s5 := Sym(112)!( 5, 32)( 6, 34)( 7, 33)( 8, 35)( 9, 37)( 10, 36)( 11, 38)
( 12, 40)( 13, 39)( 14, 50)( 15, 52)( 16, 51)( 17, 53)( 18, 55)( 19, 54)
( 20, 56)( 21, 58)( 22, 57)( 23, 41)( 24, 43)( 25, 42)( 26, 44)( 27, 46)
( 28, 45)( 29, 47)( 30, 49)( 31, 48)( 59, 86)( 60, 88)( 61, 87)( 62, 89)
( 63, 91)( 64, 90)( 65, 92)( 66, 94)( 67, 93)( 68,104)( 69,106)( 70,105)
( 71,107)( 72,109)( 73,108)( 74,110)( 75,112)( 76,111)( 77, 95)( 78, 97)
( 79, 96)( 80, 98)( 81,100)( 82, 99)( 83,101)( 84,103)( 85,102);
poly := sub<Sym(112)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s5*s5, s0*s1*s0*s1, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5,
s3*s5*s3*s5, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3,
s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3,
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;
to this polytope