Polytope of Type {2,6,12,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,12,6}*1728g
if this polytope has a name.
Group : SmallGroup(1728,47409)
Rank : 5
Schlafli Type : {2,6,12,6}
Number of vertices, edges, etc : 2, 6, 36, 36, 6
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}*864c
3-fold quotients : {2,6,4,6}*576, {2,2,12,6}*576c
4-fold quotients : {2,6,6,3}*432b
6-fold quotients : {2,2,6,6}*288b, {2,6,2,6}*288
9-fold quotients : {2,2,4,6}*192a, {2,6,4,2}*192a
12-fold quotients : {2,2,6,3}*144, {2,3,2,6}*144, {2,6,2,3}*144
18-fold quotients : {2,2,2,6}*96, {2,6,2,2}*96
24-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,2,4,2}*64
36-fold quotients : {2,2,2,3}*48, {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 := (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, 85)( 58, 84)( 59, 86)( 60, 91)( 61, 90)( 62, 92)( 63, 88)( 64, 87)
( 65, 89)( 66, 94)( 67, 93)( 68, 95)( 69,100)( 70, 99)( 71,101)( 72, 97)
( 73, 96)( 74, 98)( 75,103)( 76,102)( 77,104)( 78,109)( 79,108)( 80,110)
( 81,106)( 82,105)( 83,107);;
s3 := ( 3, 60)( 4, 61)( 5, 62)( 6, 57)( 7, 58)( 8, 59)( 9, 63)( 10, 64)
( 11, 65)( 12, 78)( 13, 79)( 14, 80)( 15, 75)( 16, 76)( 17, 77)( 18, 81)
( 19, 82)( 20, 83)( 21, 69)( 22, 70)( 23, 71)( 24, 66)( 25, 67)( 26, 68)
( 27, 72)( 28, 73)( 29, 74)( 30, 87)( 31, 88)( 32, 89)( 33, 84)( 34, 85)
( 35, 86)( 36, 90)( 37, 91)( 38, 92)( 39,105)( 40,106)( 41,107)( 42,102)
( 43,103)( 44,104)( 45,108)( 46,109)( 47,110)( 48, 96)( 49, 97)( 50, 98)
( 51, 93)( 52, 94)( 53, 95)( 54, 99)( 55,100)( 56,101);;
s4 := ( 3, 12)( 4, 13)( 5, 14)( 6, 18)( 7, 19)( 8, 20)( 9, 15)( 10, 16)
( 11, 17)( 24, 27)( 25, 28)( 26, 29)( 30, 39)( 31, 40)( 32, 41)( 33, 45)
( 34, 46)( 35, 47)( 36, 42)( 37, 43)( 38, 44)( 51, 54)( 52, 55)( 53, 56)
( 57, 66)( 58, 67)( 59, 68)( 60, 72)( 61, 73)( 62, 74)( 63, 69)( 64, 70)
( 65, 71)( 78, 81)( 79, 82)( 80, 83)( 84, 93)( 85, 94)( 86, 95)( 87, 99)
( 88,100)( 89,101)( 90, 96)( 91, 97)( 92, 98)(105,108)(106,109)(107,110);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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, 85)( 58, 84)( 59, 86)( 60, 91)( 61, 90)( 62, 92)( 63, 88)
( 64, 87)( 65, 89)( 66, 94)( 67, 93)( 68, 95)( 69,100)( 70, 99)( 71,101)
( 72, 97)( 73, 96)( 74, 98)( 75,103)( 76,102)( 77,104)( 78,109)( 79,108)
( 80,110)( 81,106)( 82,105)( 83,107);
s3 := Sym(110)!( 3, 60)( 4, 61)( 5, 62)( 6, 57)( 7, 58)( 8, 59)( 9, 63)
( 10, 64)( 11, 65)( 12, 78)( 13, 79)( 14, 80)( 15, 75)( 16, 76)( 17, 77)
( 18, 81)( 19, 82)( 20, 83)( 21, 69)( 22, 70)( 23, 71)( 24, 66)( 25, 67)
( 26, 68)( 27, 72)( 28, 73)( 29, 74)( 30, 87)( 31, 88)( 32, 89)( 33, 84)
( 34, 85)( 35, 86)( 36, 90)( 37, 91)( 38, 92)( 39,105)( 40,106)( 41,107)
( 42,102)( 43,103)( 44,104)( 45,108)( 46,109)( 47,110)( 48, 96)( 49, 97)
( 50, 98)( 51, 93)( 52, 94)( 53, 95)( 54, 99)( 55,100)( 56,101);
s4 := Sym(110)!( 3, 12)( 4, 13)( 5, 14)( 6, 18)( 7, 19)( 8, 20)( 9, 15)
( 10, 16)( 11, 17)( 24, 27)( 25, 28)( 26, 29)( 30, 39)( 31, 40)( 32, 41)
( 33, 45)( 34, 46)( 35, 47)( 36, 42)( 37, 43)( 38, 44)( 51, 54)( 52, 55)
( 53, 56)( 57, 66)( 58, 67)( 59, 68)( 60, 72)( 61, 73)( 62, 74)( 63, 69)
( 64, 70)( 65, 71)( 78, 81)( 79, 82)( 80, 83)( 84, 93)( 85, 94)( 86, 95)
( 87, 99)( 88,100)( 89,101)( 90, 96)( 91, 97)( 92, 98)(105,108)(106,109)
(107,110);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope