Polytope of Type {2,6,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6,3}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1860)
Rank : 5
Schlafli Type : {2,6,6,3}
Number of vertices, edges, etc : 2, 18, 54, 27, 3
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,3,6,3}*648b
3-fold quotients : {2,6,6,3}*432a
6-fold quotients : {2,3,6,3}*216
9-fold quotients : {2,6,2,3}*144
18-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 6, 10)( 7, 11)( 8, 9)( 12, 21)( 13, 22)( 14, 23)( 15, 28)( 16, 29)
( 17, 27)( 18, 26)( 19, 24)( 20, 25)( 33, 37)( 34, 38)( 35, 36)( 39, 48)
( 40, 49)( 41, 50)( 42, 55)( 43, 56)( 44, 54)( 45, 53)( 46, 51)( 47, 52)
( 60, 64)( 61, 65)( 62, 63)( 66, 75)( 67, 76)( 68, 77)( 69, 82)( 70, 83)
( 71, 81)( 72, 80)( 73, 78)( 74, 79)( 87, 91)( 88, 92)( 89, 90)( 93,102)
( 94,103)( 95,104)( 96,109)( 97,110)( 98,108)( 99,107)(100,105)(101,106)
(114,118)(115,119)(116,117)(120,129)(121,130)(122,131)(123,136)(124,137)
(125,135)(126,134)(127,132)(128,133)(141,145)(142,146)(143,144)(147,156)
(148,157)(149,158)(150,163)(151,164)(152,162)(153,161)(154,159)(155,160);;
s2 := ( 3, 93)( 4, 94)( 5, 95)( 6, 99)( 7,100)( 8,101)( 9, 96)( 10, 97)
( 11, 98)( 12, 84)( 13, 85)( 14, 86)( 15, 90)( 16, 91)( 17, 92)( 18, 87)
( 19, 88)( 20, 89)( 21,102)( 22,103)( 23,104)( 24,108)( 25,109)( 26,110)
( 27,105)( 28,106)( 29,107)( 30,120)( 31,121)( 32,122)( 33,126)( 34,127)
( 35,128)( 36,123)( 37,124)( 38,125)( 39,111)( 40,112)( 41,113)( 42,117)
( 43,118)( 44,119)( 45,114)( 46,115)( 47,116)( 48,129)( 49,130)( 50,131)
( 51,135)( 52,136)( 53,137)( 54,132)( 55,133)( 56,134)( 57,147)( 58,148)
( 59,149)( 60,153)( 61,154)( 62,155)( 63,150)( 64,151)( 65,152)( 66,138)
( 67,139)( 68,140)( 69,144)( 70,145)( 71,146)( 72,141)( 73,142)( 74,143)
( 75,156)( 76,157)( 77,158)( 78,162)( 79,163)( 80,164)( 81,159)( 82,160)
( 83,161);;
s3 := ( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)( 10, 35)
( 11, 34)( 12, 42)( 13, 44)( 14, 43)( 15, 39)( 16, 41)( 17, 40)( 18, 45)
( 19, 47)( 20, 46)( 21, 55)( 22, 54)( 23, 56)( 24, 52)( 25, 51)( 26, 53)
( 27, 49)( 28, 48)( 29, 50)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 69)
( 67, 71)( 68, 70)( 73, 74)( 75, 82)( 76, 81)( 77, 83)( 78, 79)( 84,111)
( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)( 92,115)
( 93,123)( 94,125)( 95,124)( 96,120)( 97,122)( 98,121)( 99,126)(100,128)
(101,127)(102,136)(103,135)(104,137)(105,133)(106,132)(107,134)(108,130)
(109,129)(110,131)(139,140)(141,144)(142,146)(143,145)(147,150)(148,152)
(149,151)(154,155)(156,163)(157,162)(158,164)(159,160);;
s4 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 57)( 31, 59)( 32, 58)( 33, 63)
( 34, 65)( 35, 64)( 36, 60)( 37, 62)( 38, 61)( 39, 66)( 40, 68)( 41, 67)
( 42, 72)( 43, 74)( 44, 73)( 45, 69)( 46, 71)( 47, 70)( 48, 75)( 49, 77)
( 50, 76)( 51, 81)( 52, 83)( 53, 82)( 54, 78)( 55, 80)( 56, 79)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,138)(112,140)(113,139)(114,144)(115,146)
(116,145)(117,141)(118,143)(119,142)(120,147)(121,149)(122,148)(123,153)
(124,155)(125,154)(126,150)(127,152)(128,151)(129,156)(130,158)(131,157)
(132,162)(133,164)(134,163)(135,159)(136,161)(137,160);;
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*s1*s0*s1,
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, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(164)!(1,2);
s1 := Sym(164)!( 6, 10)( 7, 11)( 8, 9)( 12, 21)( 13, 22)( 14, 23)( 15, 28)
( 16, 29)( 17, 27)( 18, 26)( 19, 24)( 20, 25)( 33, 37)( 34, 38)( 35, 36)
( 39, 48)( 40, 49)( 41, 50)( 42, 55)( 43, 56)( 44, 54)( 45, 53)( 46, 51)
( 47, 52)( 60, 64)( 61, 65)( 62, 63)( 66, 75)( 67, 76)( 68, 77)( 69, 82)
( 70, 83)( 71, 81)( 72, 80)( 73, 78)( 74, 79)( 87, 91)( 88, 92)( 89, 90)
( 93,102)( 94,103)( 95,104)( 96,109)( 97,110)( 98,108)( 99,107)(100,105)
(101,106)(114,118)(115,119)(116,117)(120,129)(121,130)(122,131)(123,136)
(124,137)(125,135)(126,134)(127,132)(128,133)(141,145)(142,146)(143,144)
(147,156)(148,157)(149,158)(150,163)(151,164)(152,162)(153,161)(154,159)
(155,160);
s2 := Sym(164)!( 3, 93)( 4, 94)( 5, 95)( 6, 99)( 7,100)( 8,101)( 9, 96)
( 10, 97)( 11, 98)( 12, 84)( 13, 85)( 14, 86)( 15, 90)( 16, 91)( 17, 92)
( 18, 87)( 19, 88)( 20, 89)( 21,102)( 22,103)( 23,104)( 24,108)( 25,109)
( 26,110)( 27,105)( 28,106)( 29,107)( 30,120)( 31,121)( 32,122)( 33,126)
( 34,127)( 35,128)( 36,123)( 37,124)( 38,125)( 39,111)( 40,112)( 41,113)
( 42,117)( 43,118)( 44,119)( 45,114)( 46,115)( 47,116)( 48,129)( 49,130)
( 50,131)( 51,135)( 52,136)( 53,137)( 54,132)( 55,133)( 56,134)( 57,147)
( 58,148)( 59,149)( 60,153)( 61,154)( 62,155)( 63,150)( 64,151)( 65,152)
( 66,138)( 67,139)( 68,140)( 69,144)( 70,145)( 71,146)( 72,141)( 73,142)
( 74,143)( 75,156)( 76,157)( 77,158)( 78,162)( 79,163)( 80,164)( 81,159)
( 82,160)( 83,161);
s3 := Sym(164)!( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)
( 10, 35)( 11, 34)( 12, 42)( 13, 44)( 14, 43)( 15, 39)( 16, 41)( 17, 40)
( 18, 45)( 19, 47)( 20, 46)( 21, 55)( 22, 54)( 23, 56)( 24, 52)( 25, 51)
( 26, 53)( 27, 49)( 28, 48)( 29, 50)( 58, 59)( 60, 63)( 61, 65)( 62, 64)
( 66, 69)( 67, 71)( 68, 70)( 73, 74)( 75, 82)( 76, 81)( 77, 83)( 78, 79)
( 84,111)( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)
( 92,115)( 93,123)( 94,125)( 95,124)( 96,120)( 97,122)( 98,121)( 99,126)
(100,128)(101,127)(102,136)(103,135)(104,137)(105,133)(106,132)(107,134)
(108,130)(109,129)(110,131)(139,140)(141,144)(142,146)(143,145)(147,150)
(148,152)(149,151)(154,155)(156,163)(157,162)(158,164)(159,160);
s4 := Sym(164)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 57)( 31, 59)( 32, 58)
( 33, 63)( 34, 65)( 35, 64)( 36, 60)( 37, 62)( 38, 61)( 39, 66)( 40, 68)
( 41, 67)( 42, 72)( 43, 74)( 44, 73)( 45, 69)( 46, 71)( 47, 70)( 48, 75)
( 49, 77)( 50, 76)( 51, 81)( 52, 83)( 53, 82)( 54, 78)( 55, 80)( 56, 79)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,138)(112,140)(113,139)(114,144)
(115,146)(116,145)(117,141)(118,143)(119,142)(120,147)(121,149)(122,148)
(123,153)(124,155)(125,154)(126,150)(127,152)(128,151)(129,156)(130,158)
(131,157)(132,162)(133,164)(134,163)(135,159)(136,161)(137,160);
poly := sub<Sym(164)|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*s1*s0*s1, 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,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 >;
to this polytope