Polytope of Type {4,6,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,6,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,46099)
Rank : 5
Schlafli Type : {4,6,6,4}
Number of vertices, edges, etc : 4, 18, 27, 18, 4
Order of s0s1s2s3s4 : 3
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)( 8, 16)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)
(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144);;
s1 := ( 9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 33)( 18, 34)( 19, 35)( 20, 36)
( 21, 37)( 22, 38)( 23, 39)( 24, 40)( 25, 45)( 26, 46)( 27, 47)( 28, 48)
( 29, 41)( 30, 42)( 31, 43)( 32, 44)( 49, 97)( 50, 98)( 51, 99)( 52,100)
( 53,101)( 54,102)( 55,103)( 56,104)( 57,109)( 58,110)( 59,111)( 60,112)
( 61,105)( 62,106)( 63,107)( 64,108)( 65,129)( 66,130)( 67,131)( 68,132)
( 69,133)( 70,134)( 71,135)( 72,136)( 73,141)( 74,142)( 75,143)( 76,144)
( 77,137)( 78,138)( 79,139)( 80,140)( 81,113)( 82,114)( 83,115)( 84,116)
( 85,117)( 86,118)( 87,119)( 88,120)( 89,125)( 90,126)( 91,127)( 92,128)
( 93,121)( 94,122)( 95,123)( 96,124);;
s2 := ( 1, 49)( 2, 50)( 3, 52)( 4, 51)( 5, 61)( 6, 62)( 7, 64)( 8, 63)
( 9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 53)( 14, 54)( 15, 56)( 16, 55)
( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 77)( 22, 78)( 23, 80)( 24, 79)
( 25, 73)( 26, 74)( 27, 76)( 28, 75)( 29, 69)( 30, 70)( 31, 72)( 32, 71)
( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 93)( 38, 94)( 39, 96)( 40, 95)
( 41, 89)( 42, 90)( 43, 92)( 44, 91)( 45, 85)( 46, 86)( 47, 88)( 48, 87)
( 99,100)(101,109)(102,110)(103,112)(104,111)(107,108)(115,116)(117,125)
(118,126)(119,128)(120,127)(123,124)(131,132)(133,141)(134,142)(135,144)
(136,143)(139,140);;
s3 := ( 2, 4)( 6, 8)( 10, 12)( 14, 16)( 17, 33)( 18, 36)( 19, 35)( 20, 34)
( 21, 37)( 22, 40)( 23, 39)( 24, 38)( 25, 41)( 26, 44)( 27, 43)( 28, 42)
( 29, 45)( 30, 48)( 31, 47)( 32, 46)( 49, 65)( 50, 68)( 51, 67)( 52, 66)
( 53, 69)( 54, 72)( 55, 71)( 56, 70)( 57, 73)( 58, 76)( 59, 75)( 60, 74)
( 61, 77)( 62, 80)( 63, 79)( 64, 78)( 82, 84)( 86, 88)( 90, 92)( 94, 96)
( 97,129)( 98,132)( 99,131)(100,130)(101,133)(102,136)(103,135)(104,134)
(105,137)(106,140)(107,139)(108,138)(109,141)(110,144)(111,143)(112,142)
(114,116)(118,120)(122,124)(126,128);;
s4 := ( 1, 2)( 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)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s1*s2*s3*s1*s2*s3*s1*s2*s3, s4*s3*s2*s4*s3*s4*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(144)!( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)
( 8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)
(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)
(136,144);
s1 := Sym(144)!( 9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 33)( 18, 34)( 19, 35)
( 20, 36)( 21, 37)( 22, 38)( 23, 39)( 24, 40)( 25, 45)( 26, 46)( 27, 47)
( 28, 48)( 29, 41)( 30, 42)( 31, 43)( 32, 44)( 49, 97)( 50, 98)( 51, 99)
( 52,100)( 53,101)( 54,102)( 55,103)( 56,104)( 57,109)( 58,110)( 59,111)
( 60,112)( 61,105)( 62,106)( 63,107)( 64,108)( 65,129)( 66,130)( 67,131)
( 68,132)( 69,133)( 70,134)( 71,135)( 72,136)( 73,141)( 74,142)( 75,143)
( 76,144)( 77,137)( 78,138)( 79,139)( 80,140)( 81,113)( 82,114)( 83,115)
( 84,116)( 85,117)( 86,118)( 87,119)( 88,120)( 89,125)( 90,126)( 91,127)
( 92,128)( 93,121)( 94,122)( 95,123)( 96,124);
s2 := Sym(144)!( 1, 49)( 2, 50)( 3, 52)( 4, 51)( 5, 61)( 6, 62)( 7, 64)
( 8, 63)( 9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 53)( 14, 54)( 15, 56)
( 16, 55)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 77)( 22, 78)( 23, 80)
( 24, 79)( 25, 73)( 26, 74)( 27, 76)( 28, 75)( 29, 69)( 30, 70)( 31, 72)
( 32, 71)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 93)( 38, 94)( 39, 96)
( 40, 95)( 41, 89)( 42, 90)( 43, 92)( 44, 91)( 45, 85)( 46, 86)( 47, 88)
( 48, 87)( 99,100)(101,109)(102,110)(103,112)(104,111)(107,108)(115,116)
(117,125)(118,126)(119,128)(120,127)(123,124)(131,132)(133,141)(134,142)
(135,144)(136,143)(139,140);
s3 := Sym(144)!( 2, 4)( 6, 8)( 10, 12)( 14, 16)( 17, 33)( 18, 36)( 19, 35)
( 20, 34)( 21, 37)( 22, 40)( 23, 39)( 24, 38)( 25, 41)( 26, 44)( 27, 43)
( 28, 42)( 29, 45)( 30, 48)( 31, 47)( 32, 46)( 49, 65)( 50, 68)( 51, 67)
( 52, 66)( 53, 69)( 54, 72)( 55, 71)( 56, 70)( 57, 73)( 58, 76)( 59, 75)
( 60, 74)( 61, 77)( 62, 80)( 63, 79)( 64, 78)( 82, 84)( 86, 88)( 90, 92)
( 94, 96)( 97,129)( 98,132)( 99,131)(100,130)(101,133)(102,136)(103,135)
(104,134)(105,137)(106,140)(107,139)(108,138)(109,141)(110,144)(111,143)
(112,142)(114,116)(118,120)(122,124)(126,128);
s4 := Sym(144)!( 1, 2)( 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)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144);
poly := sub<Sym(144)|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*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s1*s2*s3*s1*s2*s3*s1*s2*s3, s4*s3*s2*s4*s3*s4*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 >;
References : None.
to this polytope