Polytope of Type {2,12,6,6}

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

Permutation Representation (Magma) :

s0 := Sym(110)!(1,2);
s1 := Sym(110)!(  6,  9)(  7, 10)(  8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)
( 25, 28)( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)
( 51, 54)( 52, 55)( 53, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)
( 62, 92)( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)
( 70,100)( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)
( 78,108)( 79,109)( 80,110)( 81,105)( 82,106)( 83,107);
s2 := 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);
s3 := Sym(110)!(  3, 12)(  4, 14)(  5, 13)(  6, 18)(  7, 20)(  8, 19)(  9, 15)
( 10, 17)( 11, 16)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 39)( 31, 41)
( 32, 40)( 33, 45)( 34, 47)( 35, 46)( 36, 42)( 37, 44)( 38, 43)( 49, 50)
( 51, 54)( 52, 56)( 53, 55)( 57, 66)( 58, 68)( 59, 67)( 60, 72)( 61, 74)
( 62, 73)( 63, 69)( 64, 71)( 65, 70)( 76, 77)( 78, 81)( 79, 83)( 80, 82)
( 84, 93)( 85, 95)( 86, 94)( 87, 99)( 88,101)( 89,100)( 90, 96)( 91, 98)
( 92, 97)(103,104)(105,108)(106,110)(107,109);
s4 := Sym(110)!(  3,  4)(  6,  7)(  9, 10)( 12, 13)( 15, 16)( 18, 19)( 21, 22)
( 24, 25)( 27, 28)( 30, 31)( 33, 34)( 36, 37)( 39, 40)( 42, 43)( 45, 46)
( 48, 49)( 51, 52)( 54, 55)( 57, 58)( 60, 61)( 63, 64)( 66, 67)( 69, 70)
( 72, 73)( 75, 76)( 78, 79)( 81, 82)( 84, 85)( 87, 88)( 90, 91)( 93, 94)
( 96, 97)( 99,100)(102,103)(105,106)(108,109);
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, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s1*s2*s1*s2*s3*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 >;

to this polytope