Polytope of Type {2,4,27,2,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,27,2,2}*1728
if this polytope has a name.
Group : SmallGroup(1728,20782)
Rank : 6
Schlafli Type : {2,4,27,2,2}
Number of vertices, edges, etc : 2, 4, 54, 27, 2, 2
Order of s0s1s2s3s4s5 : 54
Order of s0s1s2s3s4s5s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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) :
3-fold quotients : {2,4,9,2,2}*576
9-fold quotients : {2,4,3,2,2}*192
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)
( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)
( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)
( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)
( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 81, 82)
( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)( 97, 98)
( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110);;
s2 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 31)( 16, 33)( 17, 32)
( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)( 25, 36)
( 26, 38)( 39, 87)( 40, 89)( 41, 88)( 42, 90)( 43, 95)( 44, 97)( 45, 96)
( 46, 98)( 47, 91)( 48, 93)( 49, 92)( 50, 94)( 51, 75)( 52, 77)( 53, 76)
( 54, 78)( 55, 83)( 56, 85)( 57, 84)( 58, 86)( 59, 79)( 60, 81)( 61, 80)
( 62, 82)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)
( 70,102)( 71,107)( 72,109)( 73,108)( 74,110);;
s3 := ( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)( 10, 49)
( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 67)( 16, 68)( 17, 70)( 18, 69)
( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 71)( 24, 72)( 25, 74)( 26, 73)
( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 51)( 32, 52)( 33, 54)( 34, 53)
( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 75, 87)( 76, 88)( 77, 90)( 78, 89)
( 79, 95)( 80, 96)( 81, 98)( 82, 97)( 83, 91)( 84, 92)( 85, 94)( 86, 93)
( 99,103)(100,104)(101,106)(102,105)(109,110);;
s4 := (111,112);;
s5 := (113,114);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;; s5 := F.6;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5,
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*s4*s3*s4, s0*s5*s0*s5,
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5,
s4*s5*s4*s5, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(114)!(1,2);
s1 := Sym(114)!( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110);
s2 := Sym(114)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 31)( 16, 33)
( 17, 32)( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)
( 25, 36)( 26, 38)( 39, 87)( 40, 89)( 41, 88)( 42, 90)( 43, 95)( 44, 97)
( 45, 96)( 46, 98)( 47, 91)( 48, 93)( 49, 92)( 50, 94)( 51, 75)( 52, 77)
( 53, 76)( 54, 78)( 55, 83)( 56, 85)( 57, 84)( 58, 86)( 59, 79)( 60, 81)
( 61, 80)( 62, 82)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)
( 69,100)( 70,102)( 71,107)( 72,109)( 73,108)( 74,110);
s3 := Sym(114)!( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)
( 10, 49)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 67)( 16, 68)( 17, 70)
( 18, 69)( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 71)( 24, 72)( 25, 74)
( 26, 73)( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 51)( 32, 52)( 33, 54)
( 34, 53)( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 75, 87)( 76, 88)( 77, 90)
( 78, 89)( 79, 95)( 80, 96)( 81, 98)( 82, 97)( 83, 91)( 84, 92)( 85, 94)
( 86, 93)( 99,103)(100,104)(101,106)(102,105)(109,110);
s4 := Sym(114)!(111,112);
s5 := Sym(114)!(113,114);
poly := sub<Sym(114)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s5*s5, 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*s4*s3*s4,
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5,
s3*s5*s3*s5, s4*s5*s4*s5, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope