Polytope of Type {4,6,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,6,3}*1152a
Also Known As : {{4,6|2},{6,6|2},{6,3}4}. if this polytope has another name.
Group : SmallGroup(1152,157559)
Rank : 5
Schlafli Type : {4,6,6,3}
Number of vertices, edges, etc : 4, 12, 24, 12, 4
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
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 : {2,6,6,3}*576
3-fold quotients : {4,2,6,3}*384
6-fold quotients : {4,2,3,3}*192, {2,2,6,3}*192
12-fold quotients : {2,2,3,3}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 73,109)( 74,110)( 75,111)( 76,112)( 77,113)( 78,114)( 79,115)( 80,116)
( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)( 86,122)( 87,123)( 88,124)
( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)( 94,130)( 95,131)( 96,132)
( 97,133)( 98,134)( 99,135)(100,136)(101,137)(102,138)(103,139)(104,140)
(105,141)(106,142)(107,143)(108,144);;
s1 := ( 1, 73)( 2, 74)( 3, 75)( 4, 76)( 5, 81)( 6, 82)( 7, 83)( 8, 84)
( 9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 93)( 18, 94)( 19, 95)( 20, 96)( 21, 89)( 22, 90)( 23, 91)( 24, 92)
( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,105)( 30,106)( 31,107)( 32,108)
( 33,101)( 34,102)( 35,103)( 36,104)( 37,109)( 38,110)( 39,111)( 40,112)
( 41,117)( 42,118)( 43,119)( 44,120)( 45,113)( 46,114)( 47,115)( 48,116)
( 49,121)( 50,122)( 51,123)( 52,124)( 53,129)( 54,130)( 55,131)( 56,132)
( 57,125)( 58,126)( 59,127)( 60,128)( 61,133)( 62,134)( 63,135)( 64,136)
( 65,141)( 66,142)( 67,143)( 68,144)( 69,137)( 70,138)( 71,139)( 72,140);;
s2 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 11, 12)( 13, 29)( 14, 30)( 15, 32)
( 16, 31)( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 21, 33)( 22, 34)( 23, 36)
( 24, 35)( 37, 41)( 38, 42)( 39, 44)( 40, 43)( 47, 48)( 49, 65)( 50, 66)
( 51, 68)( 52, 67)( 53, 61)( 54, 62)( 55, 64)( 56, 63)( 57, 69)( 58, 70)
( 59, 72)( 60, 71)( 73, 77)( 74, 78)( 75, 80)( 76, 79)( 83, 84)( 85,101)
( 86,102)( 87,104)( 88,103)( 89, 97)( 90, 98)( 91,100)( 92, 99)( 93,105)
( 94,106)( 95,108)( 96,107)(109,113)(110,114)(111,116)(112,115)(119,120)
(121,137)(122,138)(123,140)(124,139)(125,133)(126,134)(127,136)(128,135)
(129,141)(130,142)(131,144)(132,143);;
s3 := ( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 17)( 6, 20)( 7, 19)( 8, 18)
( 9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)( 37, 49)
( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)
( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73, 85)( 74, 88)
( 75, 87)( 76, 86)( 77, 89)( 78, 92)( 79, 91)( 80, 90)( 81, 93)( 82, 96)
( 83, 95)( 84, 94)( 98,100)(102,104)(106,108)(109,121)(110,124)(111,123)
(112,122)(113,125)(114,128)(115,127)(116,126)(117,129)(118,132)(119,131)
(120,130)(134,136)(138,140)(142,144);;
s4 := ( 1, 2)( 5, 6)( 9, 10)( 13, 26)( 14, 25)( 15, 27)( 16, 28)( 17, 30)
( 18, 29)( 19, 31)( 20, 32)( 21, 34)( 22, 33)( 23, 35)( 24, 36)( 37, 38)
( 41, 42)( 45, 46)( 49, 62)( 50, 61)( 51, 63)( 52, 64)( 53, 66)( 54, 65)
( 55, 67)( 56, 68)( 57, 70)( 58, 69)( 59, 71)( 60, 72)( 73, 74)( 77, 78)
( 81, 82)( 85, 98)( 86, 97)( 87, 99)( 88,100)( 89,102)( 90,101)( 91,103)
( 92,104)( 93,106)( 94,105)( 95,107)( 96,108)(109,110)(113,114)(117,118)
(121,134)(122,133)(123,135)(124,136)(125,138)(126,137)(127,139)(128,140)
(129,142)(130,141)(131,143)(132,144);;
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, s3*s4*s3*s4*s3*s4,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*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(144)!( 73,109)( 74,110)( 75,111)( 76,112)( 77,113)( 78,114)( 79,115)
( 80,116)( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)( 86,122)( 87,123)
( 88,124)( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)( 94,130)( 95,131)
( 96,132)( 97,133)( 98,134)( 99,135)(100,136)(101,137)(102,138)(103,139)
(104,140)(105,141)(106,142)(107,143)(108,144);
s1 := Sym(144)!( 1, 73)( 2, 74)( 3, 75)( 4, 76)( 5, 81)( 6, 82)( 7, 83)
( 8, 84)( 9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 85)( 14, 86)( 15, 87)
( 16, 88)( 17, 93)( 18, 94)( 19, 95)( 20, 96)( 21, 89)( 22, 90)( 23, 91)
( 24, 92)( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,105)( 30,106)( 31,107)
( 32,108)( 33,101)( 34,102)( 35,103)( 36,104)( 37,109)( 38,110)( 39,111)
( 40,112)( 41,117)( 42,118)( 43,119)( 44,120)( 45,113)( 46,114)( 47,115)
( 48,116)( 49,121)( 50,122)( 51,123)( 52,124)( 53,129)( 54,130)( 55,131)
( 56,132)( 57,125)( 58,126)( 59,127)( 60,128)( 61,133)( 62,134)( 63,135)
( 64,136)( 65,141)( 66,142)( 67,143)( 68,144)( 69,137)( 70,138)( 71,139)
( 72,140);
s2 := Sym(144)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 11, 12)( 13, 29)( 14, 30)
( 15, 32)( 16, 31)( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 21, 33)( 22, 34)
( 23, 36)( 24, 35)( 37, 41)( 38, 42)( 39, 44)( 40, 43)( 47, 48)( 49, 65)
( 50, 66)( 51, 68)( 52, 67)( 53, 61)( 54, 62)( 55, 64)( 56, 63)( 57, 69)
( 58, 70)( 59, 72)( 60, 71)( 73, 77)( 74, 78)( 75, 80)( 76, 79)( 83, 84)
( 85,101)( 86,102)( 87,104)( 88,103)( 89, 97)( 90, 98)( 91,100)( 92, 99)
( 93,105)( 94,106)( 95,108)( 96,107)(109,113)(110,114)(111,116)(112,115)
(119,120)(121,137)(122,138)(123,140)(124,139)(125,133)(126,134)(127,136)
(128,135)(129,141)(130,142)(131,144)(132,143);
s3 := Sym(144)!( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 17)( 6, 20)( 7, 19)
( 8, 18)( 9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)
( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)
( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73, 85)
( 74, 88)( 75, 87)( 76, 86)( 77, 89)( 78, 92)( 79, 91)( 80, 90)( 81, 93)
( 82, 96)( 83, 95)( 84, 94)( 98,100)(102,104)(106,108)(109,121)(110,124)
(111,123)(112,122)(113,125)(114,128)(115,127)(116,126)(117,129)(118,132)
(119,131)(120,130)(134,136)(138,140)(142,144);
s4 := Sym(144)!( 1, 2)( 5, 6)( 9, 10)( 13, 26)( 14, 25)( 15, 27)( 16, 28)
( 17, 30)( 18, 29)( 19, 31)( 20, 32)( 21, 34)( 22, 33)( 23, 35)( 24, 36)
( 37, 38)( 41, 42)( 45, 46)( 49, 62)( 50, 61)( 51, 63)( 52, 64)( 53, 66)
( 54, 65)( 55, 67)( 56, 68)( 57, 70)( 58, 69)( 59, 71)( 60, 72)( 73, 74)
( 77, 78)( 81, 82)( 85, 98)( 86, 97)( 87, 99)( 88,100)( 89,102)( 90,101)
( 91,103)( 92,104)( 93,106)( 94,105)( 95,107)( 96,108)(109,110)(113,114)
(117,118)(121,134)(122,133)(123,135)(124,136)(125,138)(126,137)(127,139)
(128,140)(129,142)(130,141)(131,143)(132,144);
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, s3*s4*s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*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 >;
References : None.
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