Polytope of Type {54,6,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,6,2}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1859)
Rank : 4
Schlafli Type : {54,6,2}
Number of vertices, edges, etc : 54, 162, 6, 2
Order of s0s1s2s3 : 54
Order of s0s1s2s3s2s1 : 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) :
3-fold quotients : {54,2,2}*432, {18,6,2}*432a
6-fold quotients : {27,2,2}*216
9-fold quotients : {18,2,2}*144, {6,6,2}*144a
18-fold quotients : {9,2,2}*72
27-fold quotients : {2,6,2}*48, {6,2,2}*48
54-fold quotients : {2,3,2}*24, {3,2,2}*24
81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 11, 12)( 13, 18)( 14, 17)( 15, 16)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 63)( 29, 62)( 30, 61)( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 72)( 38, 71)( 39, 70)( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)( 46, 81)( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)(103,108)(104,107)(105,106)(109,144)(110,143)(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,153)(119,152)(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)( 8, 32)( 9, 31)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)( 16, 51)( 17, 50)( 18, 49)( 19, 37)( 20, 39)( 21, 38)( 22, 45)( 23, 44)( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)( 90,112)( 91,127)( 92,129)( 93,128)( 94,135)( 95,134)( 96,133)( 97,132)( 98,131)( 99,130)(100,118)(101,120)(102,119)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(136,144)(137,143)(138,142)(139,141)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154);;
s2 := ( 1, 91)( 2, 92)( 3, 93)( 4, 94)( 5, 95)( 6, 96)( 7, 97)( 8, 98)( 9, 99)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)( 17, 89)( 18, 90)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)( 24,105)( 25,106)( 26,107)( 27,108)( 28,118)( 29,119)( 30,120)( 31,121)( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,109)( 38,110)( 39,111)( 40,112)( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,127)( 47,128)( 48,129)( 49,130)( 50,131)( 51,132)( 52,133)( 53,134)( 54,135)( 55,145)( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)( 64,136)( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)( 72,144)( 73,154)( 74,155)( 75,156)( 76,157)( 77,158)( 78,159)( 79,160)( 80,161)( 81,162);;
s3 := (163,164);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(164)!( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 11, 12)( 13, 18)( 14, 17)( 15, 16)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 63)( 29, 62)( 30, 61)( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 72)( 38, 71)( 39, 70)( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)( 46, 81)( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)(103,108)(104,107)(105,106)(109,144)(110,143)(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,153)(119,152)(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154);
s1 := Sym(164)!( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)( 8, 32)( 9, 31)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)( 16, 51)( 17, 50)( 18, 49)( 19, 37)( 20, 39)( 21, 38)( 22, 45)( 23, 44)( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)( 90,112)( 91,127)( 92,129)( 93,128)( 94,135)( 95,134)( 96,133)( 97,132)( 98,131)( 99,130)(100,118)(101,120)(102,119)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(136,144)(137,143)(138,142)(139,141)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154);
s2 := Sym(164)!( 1, 91)( 2, 92)( 3, 93)( 4, 94)( 5, 95)( 6, 96)( 7, 97)( 8, 98)( 9, 99)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)( 17, 89)( 18, 90)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)( 24,105)( 25,106)( 26,107)( 27,108)( 28,118)( 29,119)( 30,120)( 31,121)( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,109)( 38,110)( 39,111)( 40,112)( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,127)( 47,128)( 48,129)( 49,130)( 50,131)( 51,132)( 52,133)( 53,134)( 54,135)( 55,145)( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)( 64,136)( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)( 72,144)( 73,154)( 74,155)( 75,156)( 76,157)( 77,158)( 78,159)( 79,160)( 80,161)( 81,162);
s3 := Sym(164)!(163,164);
poly := sub<Sym(164)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
to this polytope