Polytope of Type {3,6,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,2985)
Rank : 5
Schlafli Type : {3,6,6,6}
Number of vertices, edges, etc : 3, 9, 18, 18, 6
Order of s0s1s2s3s4 : 6
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) :
3-fold quotients : {3,6,6,2}*432a, {3,2,6,6}*432a
6-fold quotients : {3,6,3,2}*216
9-fold quotients : {3,2,2,6}*144, {3,2,6,2}*144
18-fold quotients : {3,2,2,3}*72, {3,2,3,2}*72
27-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161);;
s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)( 8, 18)
( 9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)
( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)( 53, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)
( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)
( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)
(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)(116,126)
(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)(139,148)
(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)(161,162);;
s2 := ( 1, 28)( 2, 30)( 3, 29)( 4, 31)( 5, 33)( 6, 32)( 7, 34)( 8, 36)
( 9, 35)( 10, 38)( 11, 37)( 12, 39)( 13, 41)( 14, 40)( 15, 42)( 16, 44)
( 17, 43)( 18, 45)( 19, 48)( 20, 47)( 21, 46)( 22, 51)( 23, 50)( 24, 49)
( 25, 54)( 26, 53)( 27, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 65)( 67, 68)
( 70, 71)( 73, 75)( 76, 78)( 79, 81)( 82,109)( 83,111)( 84,110)( 85,112)
( 86,114)( 87,113)( 88,115)( 89,117)( 90,116)( 91,119)( 92,118)( 93,120)
( 94,122)( 95,121)( 96,123)( 97,125)( 98,124)( 99,126)(100,129)(101,128)
(102,127)(103,132)(104,131)(105,130)(106,135)(107,134)(108,133)(137,138)
(140,141)(143,144)(145,146)(148,149)(151,152)(154,156)(157,159)(160,162);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)
( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158);;
s4 := ( 1, 85)( 2, 86)( 3, 87)( 4, 82)( 5, 83)( 6, 84)( 7, 88)( 8, 89)
( 9, 90)( 10, 94)( 11, 95)( 12, 96)( 13, 91)( 14, 92)( 15, 93)( 16, 97)
( 17, 98)( 18, 99)( 19,103)( 20,104)( 21,105)( 22,100)( 23,101)( 24,102)
( 25,106)( 26,107)( 27,108)( 28,112)( 29,113)( 30,114)( 31,109)( 32,110)
( 33,111)( 34,115)( 35,116)( 36,117)( 37,121)( 38,122)( 39,123)( 40,118)
( 41,119)( 42,120)( 43,124)( 44,125)( 45,126)( 46,130)( 47,131)( 48,132)
( 49,127)( 50,128)( 51,129)( 52,133)( 53,134)( 54,135)( 55,139)( 56,140)
( 57,141)( 58,136)( 59,137)( 60,138)( 61,142)( 62,143)( 63,144)( 64,148)
( 65,149)( 66,150)( 67,145)( 68,146)( 69,147)( 70,151)( 71,152)( 72,153)
( 73,157)( 74,158)( 75,159)( 76,154)( 77,155)( 78,156)( 79,160)( 80,161)
( 81,162);;
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,
s2*s3*s4*s3*s2*s3*s4*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(162)!( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)
(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)
(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)
(153,161);
s1 := Sym(162)!( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)
( 8, 18)( 9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)
( 53, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)
( 62, 72)( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)
(107,108)(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)
(116,126)(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)
(139,148)(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)
(161,162);
s2 := Sym(162)!( 1, 28)( 2, 30)( 3, 29)( 4, 31)( 5, 33)( 6, 32)( 7, 34)
( 8, 36)( 9, 35)( 10, 38)( 11, 37)( 12, 39)( 13, 41)( 14, 40)( 15, 42)
( 16, 44)( 17, 43)( 18, 45)( 19, 48)( 20, 47)( 21, 46)( 22, 51)( 23, 50)
( 24, 49)( 25, 54)( 26, 53)( 27, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 65)
( 67, 68)( 70, 71)( 73, 75)( 76, 78)( 79, 81)( 82,109)( 83,111)( 84,110)
( 85,112)( 86,114)( 87,113)( 88,115)( 89,117)( 90,116)( 91,119)( 92,118)
( 93,120)( 94,122)( 95,121)( 96,123)( 97,125)( 98,124)( 99,126)(100,129)
(101,128)(102,127)(103,132)(104,131)(105,130)(106,135)(107,134)(108,133)
(137,138)(140,141)(143,144)(145,146)(148,149)(151,152)(154,156)(157,159)
(160,162);
s3 := Sym(162)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)
( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)
( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)
( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)
(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)
(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)
(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158);
s4 := Sym(162)!( 1, 85)( 2, 86)( 3, 87)( 4, 82)( 5, 83)( 6, 84)( 7, 88)
( 8, 89)( 9, 90)( 10, 94)( 11, 95)( 12, 96)( 13, 91)( 14, 92)( 15, 93)
( 16, 97)( 17, 98)( 18, 99)( 19,103)( 20,104)( 21,105)( 22,100)( 23,101)
( 24,102)( 25,106)( 26,107)( 27,108)( 28,112)( 29,113)( 30,114)( 31,109)
( 32,110)( 33,111)( 34,115)( 35,116)( 36,117)( 37,121)( 38,122)( 39,123)
( 40,118)( 41,119)( 42,120)( 43,124)( 44,125)( 45,126)( 46,130)( 47,131)
( 48,132)( 49,127)( 50,128)( 51,129)( 52,133)( 53,134)( 54,135)( 55,139)
( 56,140)( 57,141)( 58,136)( 59,137)( 60,138)( 61,142)( 62,143)( 63,144)
( 64,148)( 65,149)( 66,150)( 67,145)( 68,146)( 69,147)( 70,151)( 71,152)
( 72,153)( 73,157)( 74,158)( 75,159)( 76,154)( 77,155)( 78,156)( 79,160)
( 80,161)( 81,162);
poly := sub<Sym(162)|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, s2*s3*s4*s3*s2*s3*s4*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*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 >;
References : None.
to this polytope