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