Polytope of Type {2,6,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6,9}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 5
Schlafli Type : {2,6,6,9}
Number of vertices, edges, etc : 2, 6, 18, 27, 9
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 : {2,3,6,9}*648
3-fold quotients : {2,6,2,9}*432, {2,6,6,3}*432a
6-fold quotients : {2,3,2,9}*216, {2,3,6,3}*216
9-fold quotients : {2,2,2,9}*144, {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 := ( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)
( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)( 47, 55)
( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)
( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 85, 86)( 88, 89)( 91, 92)( 93,102)
( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)(101,109)
(112,113)(115,116)(118,119)(120,129)(121,131)(122,130)(123,132)(124,134)
(125,133)(126,135)(127,137)(128,136)(139,140)(142,143)(145,146)(147,156)
(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(154,164)(155,163);;
s2 := ( 3, 93)( 4, 95)( 5, 94)( 6, 96)( 7, 98)( 8, 97)( 9, 99)( 10,101)
( 11,100)( 12, 84)( 13, 86)( 14, 85)( 15, 87)( 16, 89)( 17, 88)( 18, 90)
( 19, 92)( 20, 91)( 21,102)( 22,104)( 23,103)( 24,105)( 25,107)( 26,106)
( 27,108)( 28,110)( 29,109)( 30,120)( 31,122)( 32,121)( 33,123)( 34,125)
( 35,124)( 36,126)( 37,128)( 38,127)( 39,111)( 40,113)( 41,112)( 42,114)
( 43,116)( 44,115)( 45,117)( 46,119)( 47,118)( 48,129)( 49,131)( 50,130)
( 51,132)( 52,134)( 53,133)( 54,135)( 55,137)( 56,136)( 57,147)( 58,149)
( 59,148)( 60,150)( 61,152)( 62,151)( 63,153)( 64,155)( 65,154)( 66,138)
( 67,140)( 68,139)( 69,141)( 70,143)( 71,142)( 72,144)( 73,146)( 74,145)
( 75,156)( 76,158)( 77,157)( 78,159)( 79,161)( 80,160)( 81,162)( 82,164)
( 83,163);;
s3 := ( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)( 10, 35)
( 11, 34)( 12, 40)( 13, 39)( 14, 41)( 15, 46)( 16, 45)( 17, 47)( 18, 43)
( 19, 42)( 20, 44)( 21, 50)( 22, 49)( 23, 48)( 24, 56)( 25, 55)( 26, 54)
( 27, 53)( 28, 52)( 29, 51)( 57, 63)( 58, 65)( 59, 64)( 61, 62)( 66, 73)
( 67, 72)( 68, 74)( 69, 70)( 75, 83)( 76, 82)( 77, 81)( 78, 80)( 84,111)
( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)( 92,115)
( 93,121)( 94,120)( 95,122)( 96,127)( 97,126)( 98,128)( 99,124)(100,123)
(101,125)(102,131)(103,130)(104,129)(105,137)(106,136)(107,135)(108,134)
(109,133)(110,132)(138,144)(139,146)(140,145)(142,143)(147,154)(148,153)
(149,155)(150,151)(156,164)(157,163)(158,162)(159,161);;
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, 63)( 31, 65)( 32, 64)( 33, 60)
( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)( 41, 73)
( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)( 49, 83)
( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)(115,143)
(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)(123,150)
(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)(131,163)
(132,159)(133,161)(134,160)(135,156)(136,158)(137,157);;
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*s1*s2*s3*s2*s3*s1*s2*s3*s2, 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,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(164)!(1,2);
s1 := Sym(164)!( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)
( 16, 26)( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)
( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)
( 47, 55)( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)
( 70, 80)( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 85, 86)( 88, 89)( 91, 92)
( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)
(101,109)(112,113)(115,116)(118,119)(120,129)(121,131)(122,130)(123,132)
(124,134)(125,133)(126,135)(127,137)(128,136)(139,140)(142,143)(145,146)
(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(154,164)
(155,163);
s2 := Sym(164)!( 3, 93)( 4, 95)( 5, 94)( 6, 96)( 7, 98)( 8, 97)( 9, 99)
( 10,101)( 11,100)( 12, 84)( 13, 86)( 14, 85)( 15, 87)( 16, 89)( 17, 88)
( 18, 90)( 19, 92)( 20, 91)( 21,102)( 22,104)( 23,103)( 24,105)( 25,107)
( 26,106)( 27,108)( 28,110)( 29,109)( 30,120)( 31,122)( 32,121)( 33,123)
( 34,125)( 35,124)( 36,126)( 37,128)( 38,127)( 39,111)( 40,113)( 41,112)
( 42,114)( 43,116)( 44,115)( 45,117)( 46,119)( 47,118)( 48,129)( 49,131)
( 50,130)( 51,132)( 52,134)( 53,133)( 54,135)( 55,137)( 56,136)( 57,147)
( 58,149)( 59,148)( 60,150)( 61,152)( 62,151)( 63,153)( 64,155)( 65,154)
( 66,138)( 67,140)( 68,139)( 69,141)( 70,143)( 71,142)( 72,144)( 73,146)
( 74,145)( 75,156)( 76,158)( 77,157)( 78,159)( 79,161)( 80,160)( 81,162)
( 82,164)( 83,163);
s3 := Sym(164)!( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)
( 10, 35)( 11, 34)( 12, 40)( 13, 39)( 14, 41)( 15, 46)( 16, 45)( 17, 47)
( 18, 43)( 19, 42)( 20, 44)( 21, 50)( 22, 49)( 23, 48)( 24, 56)( 25, 55)
( 26, 54)( 27, 53)( 28, 52)( 29, 51)( 57, 63)( 58, 65)( 59, 64)( 61, 62)
( 66, 73)( 67, 72)( 68, 74)( 69, 70)( 75, 83)( 76, 82)( 77, 81)( 78, 80)
( 84,111)( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)
( 92,115)( 93,121)( 94,120)( 95,122)( 96,127)( 97,126)( 98,128)( 99,124)
(100,123)(101,125)(102,131)(103,130)(104,129)(105,137)(106,136)(107,135)
(108,134)(109,133)(110,132)(138,144)(139,146)(140,145)(142,143)(147,154)
(148,153)(149,155)(150,151)(156,164)(157,163)(158,162)(159,161);
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, 63)( 31, 65)( 32, 64)
( 33, 60)( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)
( 41, 73)( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)
( 49, 83)( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)
(115,143)(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)
(123,150)(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)
(131,163)(132,159)(133,161)(134,160)(135,156)(136,158)(137,157);
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*s1*s2*s3*s2*s3*s1*s2*s3*s2,
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,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope