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