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