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