Polytope of Type {2,2,3,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,3,6,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30782)
Rank : 6
Schlafli Type : {2,2,3,6,12}
Number of vertices, edges, etc : 2, 2, 3, 9, 36, 12
Order of s0s1s2s3s4s5 : 12
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
3-fold quotients : {2,2,3,2,12}*576
4-fold quotients : {2,2,3,6,3}*432
6-fold quotients : {2,2,3,2,6}*288
9-fold quotients : {2,2,3,2,4}*192
12-fold quotients : {2,2,3,2,3}*144
18-fold quotients : {2,2,3,2,2}*96
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, 8)( 6, 10)( 7, 9)( 12, 13)( 14, 17)( 15, 19)( 16, 18)( 21, 22)
( 23, 26)( 24, 28)( 25, 27)( 30, 31)( 32, 35)( 33, 37)( 34, 36)( 39, 40)
( 41, 44)( 42, 46)( 43, 45)( 48, 49)( 50, 53)( 51, 55)( 52, 54)( 57, 58)
( 59, 62)( 60, 64)( 61, 63)( 66, 67)( 68, 71)( 69, 73)( 70, 72)( 75, 76)
( 77, 80)( 78, 82)( 79, 81)( 84, 85)( 86, 89)( 87, 91)( 88, 90)( 93, 94)
( 95, 98)( 96,100)( 97, 99)(102,103)(104,107)(105,109)(106,108)(111,112);;
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, 95)( 60, 97)( 61, 96)( 62, 99)( 63, 98)( 64,100)( 65,103)( 66,102)
( 67,101)( 68, 86)( 69, 88)( 70, 87)( 71, 90)( 72, 89)( 73, 91)( 74, 94)
( 75, 93)( 76, 92)( 77,104)( 78,106)( 79,105)( 80,108)( 81,107)( 82,109)
( 83,112)( 84,111)( 85,110);;
s5 := ( 5, 59)( 6, 61)( 7, 60)( 8, 62)( 9, 64)( 10, 63)( 11, 65)( 12, 67)
( 13, 66)( 14, 77)( 15, 79)( 16, 78)( 17, 80)( 18, 82)( 19, 81)( 20, 83)
( 21, 85)( 22, 84)( 23, 68)( 24, 70)( 25, 69)( 26, 71)( 27, 73)( 28, 72)
( 29, 74)( 30, 76)( 31, 75)( 32, 86)( 33, 88)( 34, 87)( 35, 89)( 36, 91)
( 37, 90)( 38, 92)( 39, 94)( 40, 93)( 41,104)( 42,106)( 43,105)( 44,107)
( 45,109)( 46,108)( 47,110)( 48,112)( 49,111)( 50, 95)( 51, 97)( 52, 96)
( 53, 98)( 54,100)( 55, 99)( 56,101)( 57,103)( 58,102);;
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,
s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3,
s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*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, 8)( 6, 10)( 7, 9)( 12, 13)( 14, 17)( 15, 19)( 16, 18)
( 21, 22)( 23, 26)( 24, 28)( 25, 27)( 30, 31)( 32, 35)( 33, 37)( 34, 36)
( 39, 40)( 41, 44)( 42, 46)( 43, 45)( 48, 49)( 50, 53)( 51, 55)( 52, 54)
( 57, 58)( 59, 62)( 60, 64)( 61, 63)( 66, 67)( 68, 71)( 69, 73)( 70, 72)
( 75, 76)( 77, 80)( 78, 82)( 79, 81)( 84, 85)( 86, 89)( 87, 91)( 88, 90)
( 93, 94)( 95, 98)( 96,100)( 97, 99)(102,103)(104,107)(105,109)(106,108)
(111,112);
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, 95)( 60, 97)( 61, 96)( 62, 99)( 63, 98)( 64,100)( 65,103)
( 66,102)( 67,101)( 68, 86)( 69, 88)( 70, 87)( 71, 90)( 72, 89)( 73, 91)
( 74, 94)( 75, 93)( 76, 92)( 77,104)( 78,106)( 79,105)( 80,108)( 81,107)
( 82,109)( 83,112)( 84,111)( 85,110);
s5 := Sym(112)!( 5, 59)( 6, 61)( 7, 60)( 8, 62)( 9, 64)( 10, 63)( 11, 65)
( 12, 67)( 13, 66)( 14, 77)( 15, 79)( 16, 78)( 17, 80)( 18, 82)( 19, 81)
( 20, 83)( 21, 85)( 22, 84)( 23, 68)( 24, 70)( 25, 69)( 26, 71)( 27, 73)
( 28, 72)( 29, 74)( 30, 76)( 31, 75)( 32, 86)( 33, 88)( 34, 87)( 35, 89)
( 36, 91)( 37, 90)( 38, 92)( 39, 94)( 40, 93)( 41,104)( 42,106)( 43,105)
( 44,107)( 45,109)( 46,108)( 47,110)( 48,112)( 49,111)( 50, 95)( 51, 97)
( 52, 96)( 53, 98)( 54,100)( 55, 99)( 56,101)( 57,103)( 58,102);
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, s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3,
s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;
to this polytope