Polytope of Type {6,6,3,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,3,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,2985)
Rank : 5
Schlafli Type : {6,6,3,6}
Number of vertices, edges, etc : 6, 18, 9, 9, 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) :
2-fold quotients : {3,6,3,6}*648
3-fold quotients : {6,6,3,2}*432a, {6,2,3,6}*432
6-fold quotients : {3,6,3,2}*216, {3,2,3,6}*216
9-fold quotients : {2,2,3,6}*144, {6,2,3,2}*144
18-fold quotients : {3,2,3,2}*72
27-fold quotients : {2,2,3,2}*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, 91)( 2, 93)( 3, 92)( 4, 94)( 5, 96)( 6, 95)( 7, 97)( 8, 99)
( 9, 98)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)( 16, 88)
( 17, 90)( 18, 89)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)( 24,104)
( 25,106)( 26,108)( 27,107)( 28,118)( 29,120)( 30,119)( 31,121)( 32,123)
( 33,122)( 34,124)( 35,126)( 36,125)( 37,109)( 38,111)( 39,110)( 40,112)
( 41,114)( 42,113)( 43,115)( 44,117)( 45,116)( 46,127)( 47,129)( 48,128)
( 49,130)( 50,132)( 51,131)( 52,133)( 53,135)( 54,134)( 55,145)( 56,147)
( 57,146)( 58,148)( 59,150)( 60,149)( 61,151)( 62,153)( 63,152)( 64,136)
( 65,138)( 66,137)( 67,139)( 68,141)( 69,140)( 70,142)( 71,144)( 72,143)
( 73,154)( 74,156)( 75,155)( 76,157)( 77,159)( 78,158)( 79,160)( 80,162)
( 81,161);;
s2 := ( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)( 8, 33)
( 9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)( 16, 41)
( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)( 24, 52)
( 25, 51)( 26, 50)( 27, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 65)
( 67, 71)( 68, 70)( 69, 72)( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82,109)
( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)( 90,113)
( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)( 98,121)
( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)(106,132)
(107,131)(108,130)(137,138)(139,142)(140,144)(141,143)(145,146)(148,152)
(149,151)(150,153)(154,156)(157,162)(158,161)(159,160);;
s3 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 58)( 29, 60)( 30, 59)( 31, 55)
( 32, 57)( 33, 56)( 34, 61)( 35, 63)( 36, 62)( 37, 67)( 38, 69)( 39, 68)
( 40, 64)( 41, 66)( 42, 65)( 43, 70)( 44, 72)( 45, 71)( 46, 76)( 47, 78)
( 48, 77)( 49, 73)( 50, 75)( 51, 74)( 52, 79)( 53, 81)( 54, 80)( 82, 85)
( 83, 87)( 84, 86)( 89, 90)( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)
(101,105)(102,104)(107,108)(109,139)(110,141)(111,140)(112,136)(113,138)
(114,137)(115,142)(116,144)(117,143)(118,148)(119,150)(120,149)(121,145)
(122,147)(123,146)(124,151)(125,153)(126,152)(127,157)(128,159)(129,158)
(130,154)(131,156)(132,155)(133,160)(134,162)(135,161);;
s4 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)
(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162);;
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, s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
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, 91)( 2, 93)( 3, 92)( 4, 94)( 5, 96)( 6, 95)( 7, 97)
( 8, 99)( 9, 98)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)
( 16, 88)( 17, 90)( 18, 89)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)
( 24,104)( 25,106)( 26,108)( 27,107)( 28,118)( 29,120)( 30,119)( 31,121)
( 32,123)( 33,122)( 34,124)( 35,126)( 36,125)( 37,109)( 38,111)( 39,110)
( 40,112)( 41,114)( 42,113)( 43,115)( 44,117)( 45,116)( 46,127)( 47,129)
( 48,128)( 49,130)( 50,132)( 51,131)( 52,133)( 53,135)( 54,134)( 55,145)
( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,151)( 62,153)( 63,152)
( 64,136)( 65,138)( 66,137)( 67,139)( 68,141)( 69,140)( 70,142)( 71,144)
( 72,143)( 73,154)( 74,156)( 75,155)( 76,157)( 77,159)( 78,158)( 79,160)
( 80,162)( 81,161);
s2 := Sym(162)!( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)
( 8, 33)( 9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)
( 16, 41)( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)
( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)
( 64, 65)( 67, 71)( 68, 70)( 69, 72)( 73, 75)( 76, 81)( 77, 80)( 78, 79)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)
( 98,121)( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)
(106,132)(107,131)(108,130)(137,138)(139,142)(140,144)(141,143)(145,146)
(148,152)(149,151)(150,153)(154,156)(157,162)(158,161)(159,160);
s3 := Sym(162)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 13)( 11, 15)( 12, 14)
( 17, 18)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 58)( 29, 60)( 30, 59)
( 31, 55)( 32, 57)( 33, 56)( 34, 61)( 35, 63)( 36, 62)( 37, 67)( 38, 69)
( 39, 68)( 40, 64)( 41, 66)( 42, 65)( 43, 70)( 44, 72)( 45, 71)( 46, 76)
( 47, 78)( 48, 77)( 49, 73)( 50, 75)( 51, 74)( 52, 79)( 53, 81)( 54, 80)
( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91, 94)( 92, 96)( 93, 95)( 98, 99)
(100,103)(101,105)(102,104)(107,108)(109,139)(110,141)(111,140)(112,136)
(113,138)(114,137)(115,142)(116,144)(117,143)(118,148)(119,150)(120,149)
(121,145)(122,147)(123,146)(124,151)(125,153)(126,152)(127,157)(128,159)
(129,158)(130,154)(131,156)(132,155)(133,160)(134,162)(135,161);
s4 := Sym(162)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)
(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)
(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162);
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, s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope