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