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

to this polytope