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}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30804)
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}*864a
3-fold quotients : {2,6,4,6}*576
4-fold quotients : {2,3,6,6}*432a, {2,6,6,3}*432a
6-fold quotients : {2,6,2,6}*288
8-fold quotients : {2,3,6,3}*216
9-fold quotients : {2,2,4,6}*192a, {2,6,4,2}*192a
12-fold quotients : {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)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)( 71, 73)
( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)( 89, 91)
( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109);;
s2 := ( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 15)( 13, 17)( 14, 16)( 19, 20)
( 21, 24)( 22, 26)( 23, 25)( 28, 29)( 30, 33)( 31, 35)( 32, 34)( 37, 38)
( 39, 42)( 40, 44)( 41, 43)( 46, 47)( 48, 51)( 49, 53)( 50, 52)( 55, 56)
( 57, 87)( 58, 89)( 59, 88)( 60, 84)( 61, 86)( 62, 85)( 63, 90)( 64, 92)
( 65, 91)( 66, 96)( 67, 98)( 68, 97)( 69, 93)( 70, 95)( 71, 94)( 72, 99)
( 73,101)( 74,100)( 75,105)( 76,107)( 77,106)( 78,102)( 79,104)( 80,103)
( 81,108)( 82,110)( 83,109);;
s3 := ( 3, 66)( 4, 68)( 5, 67)( 6, 70)( 7, 69)( 8, 71)( 9, 74)( 10, 73)
( 11, 72)( 12, 57)( 13, 59)( 14, 58)( 15, 61)( 16, 60)( 17, 62)( 18, 65)
( 19, 64)( 20, 63)( 21, 75)( 22, 77)( 23, 76)( 24, 79)( 25, 78)( 26, 80)
( 27, 83)( 28, 82)( 29, 81)( 30, 93)( 31, 95)( 32, 94)( 33, 97)( 34, 96)
( 35, 98)( 36,101)( 37,100)( 38, 99)( 39, 84)( 40, 86)( 41, 85)( 42, 88)
( 43, 87)( 44, 89)( 45, 92)( 46, 91)( 47, 90)( 48,102)( 49,104)( 50,103)
( 51,106)( 52,105)( 53,107)( 54,110)( 55,109)( 56,108);;
s4 := ( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)
( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)( 47, 55)
( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)
( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 85, 86)( 88, 89)( 91, 92)( 93,102)
( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)(101,109);;
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,
s3*s1*s2*s3*s2*s3*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)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)
( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)
( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)
(107,109);
s2 := Sym(110)!( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 15)( 13, 17)( 14, 16)
( 19, 20)( 21, 24)( 22, 26)( 23, 25)( 28, 29)( 30, 33)( 31, 35)( 32, 34)
( 37, 38)( 39, 42)( 40, 44)( 41, 43)( 46, 47)( 48, 51)( 49, 53)( 50, 52)
( 55, 56)( 57, 87)( 58, 89)( 59, 88)( 60, 84)( 61, 86)( 62, 85)( 63, 90)
( 64, 92)( 65, 91)( 66, 96)( 67, 98)( 68, 97)( 69, 93)( 70, 95)( 71, 94)
( 72, 99)( 73,101)( 74,100)( 75,105)( 76,107)( 77,106)( 78,102)( 79,104)
( 80,103)( 81,108)( 82,110)( 83,109);
s3 := Sym(110)!( 3, 66)( 4, 68)( 5, 67)( 6, 70)( 7, 69)( 8, 71)( 9, 74)
( 10, 73)( 11, 72)( 12, 57)( 13, 59)( 14, 58)( 15, 61)( 16, 60)( 17, 62)
( 18, 65)( 19, 64)( 20, 63)( 21, 75)( 22, 77)( 23, 76)( 24, 79)( 25, 78)
( 26, 80)( 27, 83)( 28, 82)( 29, 81)( 30, 93)( 31, 95)( 32, 94)( 33, 97)
( 34, 96)( 35, 98)( 36,101)( 37,100)( 38, 99)( 39, 84)( 40, 86)( 41, 85)
( 42, 88)( 43, 87)( 44, 89)( 45, 92)( 46, 91)( 47, 90)( 48,102)( 49,104)
( 50,103)( 51,106)( 52,105)( 53,107)( 54,110)( 55,109)( 56,108);
s4 := Sym(110)!( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)
( 16, 26)( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)
( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)
( 47, 55)( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)
( 70, 80)( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 85, 86)( 88, 89)( 91, 92)
( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)
(101,109);
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, s3*s1*s2*s3*s2*s3*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