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}*1296b
if this polytope has a name.
Group : SmallGroup(1296,1860)
Rank : 5
Schlafli Type : {2,6,6,3}
Number of vertices, edges, etc : 2, 6, 54, 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 : {2,3,6,3}*648a
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 := ( 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);;
s2 := ( 3,111)( 4,113)( 5,112)( 6,117)( 7,119)( 8,118)( 9,114)( 10,116)
( 11,115)( 12,123)( 13,125)( 14,124)( 15,120)( 16,122)( 17,121)( 18,126)
( 19,128)( 20,127)( 21,136)( 22,135)( 23,137)( 24,133)( 25,132)( 26,134)
( 27,130)( 28,129)( 29,131)( 30, 84)( 31, 86)( 32, 85)( 33, 90)( 34, 92)
( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39, 96)( 40, 98)( 41, 97)( 42, 93)
( 43, 95)( 44, 94)( 45, 99)( 46,101)( 47,100)( 48,109)( 49,108)( 50,110)
( 51,106)( 52,105)( 53,107)( 54,103)( 55,102)( 56,104)( 57,138)( 58,140)
( 59,139)( 60,144)( 61,146)( 62,145)( 63,141)( 64,143)( 65,142)( 66,150)
( 67,152)( 68,151)( 69,147)( 70,149)( 71,148)( 72,153)( 73,155)( 74,154)
( 75,163)( 76,162)( 77,164)( 78,160)( 79,159)( 80,161)( 81,157)( 82,156)
( 83,158);;
s3 := ( 3, 12)( 4, 13)( 5, 14)( 6, 18)( 7, 19)( 8, 20)( 9, 15)( 10, 16)
( 11, 17)( 24, 27)( 25, 28)( 26, 29)( 30, 39)( 31, 40)( 32, 41)( 33, 45)
( 34, 46)( 35, 47)( 36, 42)( 37, 43)( 38, 44)( 51, 54)( 52, 55)( 53, 56)
( 57, 66)( 58, 67)( 59, 68)( 60, 72)( 61, 73)( 62, 74)( 63, 69)( 64, 70)
( 65, 71)( 78, 81)( 79, 82)( 80, 83)( 84, 93)( 85, 94)( 86, 95)( 87, 99)
( 88,100)( 89,101)( 90, 96)( 91, 97)( 92, 98)(105,108)(106,109)(107,110)
(111,120)(112,121)(113,122)(114,126)(115,127)(116,128)(117,123)(118,124)
(119,125)(132,135)(133,136)(134,137)(138,147)(139,148)(140,149)(141,153)
(142,154)(143,155)(144,150)(145,151)(146,152)(159,162)(160,163)(161,164);;
s4 := ( 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);;
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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
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)!( 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);
s2 := Sym(164)!( 3,111)( 4,113)( 5,112)( 6,117)( 7,119)( 8,118)( 9,114)
( 10,116)( 11,115)( 12,123)( 13,125)( 14,124)( 15,120)( 16,122)( 17,121)
( 18,126)( 19,128)( 20,127)( 21,136)( 22,135)( 23,137)( 24,133)( 25,132)
( 26,134)( 27,130)( 28,129)( 29,131)( 30, 84)( 31, 86)( 32, 85)( 33, 90)
( 34, 92)( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39, 96)( 40, 98)( 41, 97)
( 42, 93)( 43, 95)( 44, 94)( 45, 99)( 46,101)( 47,100)( 48,109)( 49,108)
( 50,110)( 51,106)( 52,105)( 53,107)( 54,103)( 55,102)( 56,104)( 57,138)
( 58,140)( 59,139)( 60,144)( 61,146)( 62,145)( 63,141)( 64,143)( 65,142)
( 66,150)( 67,152)( 68,151)( 69,147)( 70,149)( 71,148)( 72,153)( 73,155)
( 74,154)( 75,163)( 76,162)( 77,164)( 78,160)( 79,159)( 80,161)( 81,157)
( 82,156)( 83,158);
s3 := Sym(164)!( 3, 12)( 4, 13)( 5, 14)( 6, 18)( 7, 19)( 8, 20)( 9, 15)
( 10, 16)( 11, 17)( 24, 27)( 25, 28)( 26, 29)( 30, 39)( 31, 40)( 32, 41)
( 33, 45)( 34, 46)( 35, 47)( 36, 42)( 37, 43)( 38, 44)( 51, 54)( 52, 55)
( 53, 56)( 57, 66)( 58, 67)( 59, 68)( 60, 72)( 61, 73)( 62, 74)( 63, 69)
( 64, 70)( 65, 71)( 78, 81)( 79, 82)( 80, 83)( 84, 93)( 85, 94)( 86, 95)
( 87, 99)( 88,100)( 89,101)( 90, 96)( 91, 97)( 92, 98)(105,108)(106,109)
(107,110)(111,120)(112,121)(113,122)(114,126)(115,127)(116,128)(117,123)
(118,124)(119,125)(132,135)(133,136)(134,137)(138,147)(139,148)(140,149)
(141,153)(142,154)(143,155)(144,150)(145,151)(146,152)(159,162)(160,163)
(161,164);
s4 := 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);
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,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 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