Polytope of Type {2,6,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6,12}*1728d
if this polytope has a name.
Group : SmallGroup(1728,47319)
Rank : 5
Schlafli Type : {2,6,6,12}
Number of vertices, edges, etc : 2, 6, 18, 36, 12
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 : {2,6,6,6}*864d
3-fold quotients : {2,2,6,12}*576a, {2,6,6,4}*576b
6-fold quotients : {2,2,6,6}*288a, {2,6,6,2}*288b
9-fold quotients : {2,2,2,12}*192, {2,2,6,4}*192a
12-fold quotients : {2,6,3,2}*144
18-fold quotients : {2,2,2,6}*96, {2,2,6,2}*96
27-fold quotients : {2,2,2,4}*64
36-fold quotients : {2,2,2,3}*48, {2,2,3,2}*48
54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)( 22, 23)( 25, 26)
( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 40, 41)( 43, 44)( 46, 47)( 49, 50)
( 52, 53)( 55, 56)( 58, 59)( 61, 62)( 64, 65)( 67, 68)( 70, 71)( 73, 74)
( 76, 77)( 79, 80)( 82, 83)( 85, 86)( 88, 89)( 91, 92)( 94, 95)( 97, 98)
(100,101)(103,104)(106,107)(109,110);;
s2 := ( 3, 4)( 6, 10)( 7, 9)( 8, 11)( 12, 13)( 15, 19)( 16, 18)( 17, 20)
( 21, 22)( 24, 28)( 25, 27)( 26, 29)( 30, 31)( 33, 37)( 34, 36)( 35, 38)
( 39, 40)( 42, 46)( 43, 45)( 44, 47)( 48, 49)( 51, 55)( 52, 54)( 53, 56)
( 57, 58)( 60, 64)( 61, 63)( 62, 65)( 66, 67)( 69, 73)( 70, 72)( 71, 74)
( 75, 76)( 78, 82)( 79, 81)( 80, 83)( 84, 85)( 87, 91)( 88, 90)( 89, 92)
( 93, 94)( 96,100)( 97, 99)( 98,101)(102,103)(105,109)(106,108)(107,110);;
s3 := ( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 21)
( 16, 23)( 17, 22)( 18, 27)( 19, 29)( 20, 28)( 30, 33)( 31, 35)( 32, 34)
( 37, 38)( 39, 51)( 40, 53)( 41, 52)( 42, 48)( 43, 50)( 44, 49)( 45, 54)
( 46, 56)( 47, 55)( 57, 87)( 58, 89)( 59, 88)( 60, 84)( 61, 86)( 62, 85)
( 63, 90)( 64, 92)( 65, 91)( 66,105)( 67,107)( 68,106)( 69,102)( 70,104)
( 71,103)( 72,108)( 73,110)( 74,109)( 75, 96)( 76, 98)( 77, 97)( 78, 93)
( 79, 95)( 80, 94)( 81, 99)( 82,101)( 83,100);;
s4 := ( 3, 66)( 4, 67)( 5, 68)( 6, 69)( 7, 70)( 8, 71)( 9, 72)( 10, 73)
( 11, 74)( 12, 57)( 13, 58)( 14, 59)( 15, 60)( 16, 61)( 17, 62)( 18, 63)
( 19, 64)( 20, 65)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)
( 27, 81)( 28, 82)( 29, 83)( 30, 93)( 31, 94)( 32, 95)( 33, 96)( 34, 97)
( 35, 98)( 36, 99)( 37,100)( 38,101)( 39, 84)( 40, 85)( 41, 86)( 42, 87)
( 43, 88)( 44, 89)( 45, 90)( 46, 91)( 47, 92)( 48,102)( 49,103)( 50,104)
( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(110)!(1,2);
s1 := Sym(110)!( 4, 5)( 7, 8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)( 22, 23)
( 25, 26)( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 40, 41)( 43, 44)( 46, 47)
( 49, 50)( 52, 53)( 55, 56)( 58, 59)( 61, 62)( 64, 65)( 67, 68)( 70, 71)
( 73, 74)( 76, 77)( 79, 80)( 82, 83)( 85, 86)( 88, 89)( 91, 92)( 94, 95)
( 97, 98)(100,101)(103,104)(106,107)(109,110);
s2 := Sym(110)!( 3, 4)( 6, 10)( 7, 9)( 8, 11)( 12, 13)( 15, 19)( 16, 18)
( 17, 20)( 21, 22)( 24, 28)( 25, 27)( 26, 29)( 30, 31)( 33, 37)( 34, 36)
( 35, 38)( 39, 40)( 42, 46)( 43, 45)( 44, 47)( 48, 49)( 51, 55)( 52, 54)
( 53, 56)( 57, 58)( 60, 64)( 61, 63)( 62, 65)( 66, 67)( 69, 73)( 70, 72)
( 71, 74)( 75, 76)( 78, 82)( 79, 81)( 80, 83)( 84, 85)( 87, 91)( 88, 90)
( 89, 92)( 93, 94)( 96,100)( 97, 99)( 98,101)(102,103)(105,109)(106,108)
(107,110);
s3 := Sym(110)!( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 24)( 13, 26)( 14, 25)
( 15, 21)( 16, 23)( 17, 22)( 18, 27)( 19, 29)( 20, 28)( 30, 33)( 31, 35)
( 32, 34)( 37, 38)( 39, 51)( 40, 53)( 41, 52)( 42, 48)( 43, 50)( 44, 49)
( 45, 54)( 46, 56)( 47, 55)( 57, 87)( 58, 89)( 59, 88)( 60, 84)( 61, 86)
( 62, 85)( 63, 90)( 64, 92)( 65, 91)( 66,105)( 67,107)( 68,106)( 69,102)
( 70,104)( 71,103)( 72,108)( 73,110)( 74,109)( 75, 96)( 76, 98)( 77, 97)
( 78, 93)( 79, 95)( 80, 94)( 81, 99)( 82,101)( 83,100);
s4 := Sym(110)!( 3, 66)( 4, 67)( 5, 68)( 6, 69)( 7, 70)( 8, 71)( 9, 72)
( 10, 73)( 11, 74)( 12, 57)( 13, 58)( 14, 59)( 15, 60)( 16, 61)( 17, 62)
( 18, 63)( 19, 64)( 20, 65)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)
( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 93)( 31, 94)( 32, 95)( 33, 96)
( 34, 97)( 35, 98)( 36, 99)( 37,100)( 38,101)( 39, 84)( 40, 85)( 41, 86)
( 42, 87)( 43, 88)( 44, 89)( 45, 90)( 46, 91)( 47, 92)( 48,102)( 49,103)
( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110);
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*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope