Polytope of Type {2,3,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1860)
Rank : 5
Schlafli Type : {2,3,6,6}
Number of vertices, edges, etc : 2, 9, 27, 54, 6
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 : {2,3,6,6}*432a
6-fold quotients : {2,3,6,3}*216
9-fold quotients : {2,3,2,6}*144
18-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,3,2,2}*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, 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);;
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 := ( 3, 84)( 4, 86)( 5, 85)( 6, 90)( 7, 92)( 8, 91)( 9, 87)( 10, 89)
( 11, 88)( 12, 93)( 13, 95)( 14, 94)( 15, 99)( 16,101)( 17,100)( 18, 96)
( 19, 98)( 20, 97)( 21,102)( 22,104)( 23,103)( 24,108)( 25,110)( 26,109)
( 27,105)( 28,107)( 29,106)( 30,138)( 31,140)( 32,139)( 33,144)( 34,146)
( 35,145)( 36,141)( 37,143)( 38,142)( 39,147)( 40,149)( 41,148)( 42,153)
( 43,155)( 44,154)( 45,150)( 46,152)( 47,151)( 48,156)( 49,158)( 50,157)
( 51,162)( 52,164)( 53,163)( 54,159)( 55,161)( 56,160)( 57,111)( 58,113)
( 59,112)( 60,117)( 61,119)( 62,118)( 63,114)( 64,116)( 65,115)( 66,120)
( 67,122)( 68,121)( 69,126)( 70,128)( 71,127)( 72,123)( 73,125)( 74,124)
( 75,129)( 76,131)( 77,130)( 78,135)( 79,137)( 80,136)( 81,132)( 82,134)
( 83,133);;
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,
s1*s2*s1*s2*s1*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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)!( 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, 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);
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)!( 3, 84)( 4, 86)( 5, 85)( 6, 90)( 7, 92)( 8, 91)( 9, 87)
( 10, 89)( 11, 88)( 12, 93)( 13, 95)( 14, 94)( 15, 99)( 16,101)( 17,100)
( 18, 96)( 19, 98)( 20, 97)( 21,102)( 22,104)( 23,103)( 24,108)( 25,110)
( 26,109)( 27,105)( 28,107)( 29,106)( 30,138)( 31,140)( 32,139)( 33,144)
( 34,146)( 35,145)( 36,141)( 37,143)( 38,142)( 39,147)( 40,149)( 41,148)
( 42,153)( 43,155)( 44,154)( 45,150)( 46,152)( 47,151)( 48,156)( 49,158)
( 50,157)( 51,162)( 52,164)( 53,163)( 54,159)( 55,161)( 56,160)( 57,111)
( 58,113)( 59,112)( 60,117)( 61,119)( 62,118)( 63,114)( 64,116)( 65,115)
( 66,120)( 67,122)( 68,121)( 69,126)( 70,128)( 71,127)( 72,123)( 73,125)
( 74,124)( 75,129)( 76,131)( 77,130)( 78,135)( 79,137)( 80,136)( 81,132)
( 82,134)( 83,133);
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, s1*s2*s1*s2*s1*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope