Polytope of Type {12,6,6,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,6,2}*1728g
if this polytope has a name.
Group : SmallGroup(1728,47409)
Rank : 5
Schlafli Type : {12,6,6,2}
Number of vertices, edges, etc : 12, 36, 18, 6, 2
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 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 : {6,6,6,2}*864f
3-fold quotients : {4,6,6,2}*576b, {12,6,2,2}*576c
4-fold quotients : {6,3,6,2}*432
6-fold quotients : {2,6,6,2}*288c, {6,6,2,2}*288b
9-fold quotients : {4,6,2,2}*192a
12-fold quotients : {2,3,6,2}*144, {6,3,2,2}*144
18-fold quotients : {2,6,2,2}*96
27-fold quotients : {4,2,2,2}*64
36-fold quotients : {2,3,2,2}*48
54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105);;
s1 := ( 1, 58)( 2, 60)( 3, 59)( 4, 55)( 5, 57)( 6, 56)( 7, 61)( 8, 63)
( 9, 62)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)( 16, 79)
( 17, 81)( 18, 80)( 19, 67)( 20, 69)( 21, 68)( 22, 64)( 23, 66)( 24, 65)
( 25, 70)( 26, 72)( 27, 71)( 28, 85)( 29, 87)( 30, 86)( 31, 82)( 32, 84)
( 33, 83)( 34, 88)( 35, 90)( 36, 89)( 37,103)( 38,105)( 39,104)( 40,100)
( 41,102)( 42,101)( 43,106)( 44,108)( 45,107)( 46, 94)( 47, 96)( 48, 95)
( 49, 91)( 50, 93)( 51, 92)( 52, 97)( 53, 99)( 54, 98);;
s2 := ( 1, 11)( 2, 10)( 3, 12)( 4, 17)( 5, 16)( 6, 18)( 7, 14)( 8, 13)
( 9, 15)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 38)( 29, 37)( 30, 39)
( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 47)( 49, 53)
( 50, 52)( 51, 54)( 55, 65)( 56, 64)( 57, 66)( 58, 71)( 59, 70)( 60, 72)
( 61, 68)( 62, 67)( 63, 69)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82, 92)
( 83, 91)( 84, 93)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)( 90, 96)
(100,101)(103,107)(104,106)(105,108);;
s3 := ( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)
( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)( 71, 72)
( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)( 95, 96)
( 98, 99)(101,102)(104,105)(107,108);;
s4 := (109,110);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(110)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)
( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)
( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)
( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105);
s1 := Sym(110)!( 1, 58)( 2, 60)( 3, 59)( 4, 55)( 5, 57)( 6, 56)( 7, 61)
( 8, 63)( 9, 62)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)
( 16, 79)( 17, 81)( 18, 80)( 19, 67)( 20, 69)( 21, 68)( 22, 64)( 23, 66)
( 24, 65)( 25, 70)( 26, 72)( 27, 71)( 28, 85)( 29, 87)( 30, 86)( 31, 82)
( 32, 84)( 33, 83)( 34, 88)( 35, 90)( 36, 89)( 37,103)( 38,105)( 39,104)
( 40,100)( 41,102)( 42,101)( 43,106)( 44,108)( 45,107)( 46, 94)( 47, 96)
( 48, 95)( 49, 91)( 50, 93)( 51, 92)( 52, 97)( 53, 99)( 54, 98);
s2 := Sym(110)!( 1, 11)( 2, 10)( 3, 12)( 4, 17)( 5, 16)( 6, 18)( 7, 14)
( 8, 13)( 9, 15)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 38)( 29, 37)
( 30, 39)( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 47)
( 49, 53)( 50, 52)( 51, 54)( 55, 65)( 56, 64)( 57, 66)( 58, 71)( 59, 70)
( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 74)( 76, 80)( 77, 79)( 78, 81)
( 82, 92)( 83, 91)( 84, 93)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)
( 90, 96)(100,101)(103,107)(104,106)(105,108);
s3 := Sym(110)!( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)
( 47, 48)( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)
( 71, 72)( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)
( 95, 96)( 98, 99)(101,102)(104,105)(107,108);
s4 := Sym(110)!(109,110);
poly := sub<Sym(110)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 >;
to this polytope