Polytope of Type {3,6,6,6} with group SmallGroup(1296,3585)

Overview

Group: SmallGroup(1296,3585)
Rank: 5
Schläfli Type: {3,6,6,6}
Vertices, edges, …: 3, 9, 18, 18, 6
Order of s₀s₁s₂s₃s₄: 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{3,6,6,3}*648

3-fold

6-fold

9-fold

18-fold

{3,2,2,3}*72

27-fold

{3,2,2,2}*48

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)( 34, 58)( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)( 42, 72)( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)( 50, 80)( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159);;
s1 := (  1, 31)(  2, 32)(  3, 33)(  4, 28)(  5, 29)(  6, 30)(  7, 34)(  8, 35)(  9, 36)( 10, 40)( 11, 41)( 12, 42)( 13, 37)( 14, 38)( 15, 39)( 16, 43)( 17, 44)( 18, 45)( 19, 49)( 20, 50)( 21, 51)( 22, 46)( 23, 47)( 24, 48)( 25, 52)( 26, 53)( 27, 54)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)( 73, 76)( 74, 77)( 75, 78)( 82,112)( 83,113)( 84,114)( 85,109)( 86,110)( 87,111)( 88,115)( 89,116)( 90,117)( 91,121)( 92,122)( 93,123)( 94,118)( 95,119)( 96,120)( 97,124)( 98,125)( 99,126)(100,130)(101,131)(102,132)(103,127)(104,128)(105,129)(106,133)(107,134)(108,135)(136,139)(137,140)(138,141)(145,148)(146,149)(147,150)(154,157)(155,158)(156,159);;
s2 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)( 18, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)( 36, 63)( 37, 73)( 38, 74)( 39, 75)( 40, 76)( 41, 77)( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 64)( 47, 65)( 48, 66)( 49, 67)( 50, 68)( 51, 69)( 52, 70)( 53, 71)( 54, 72)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)(119,155)(120,156)(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153);;
s3 := (  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)(  9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)( 53, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)(116,126)(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)(139,148)(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)(161,162);;
s4 := (  1, 83)(  2, 82)(  3, 84)(  4, 86)(  5, 85)(  6, 87)(  7, 89)(  8, 88)(  9, 90)( 10,101)( 11,100)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)( 18,108)( 19, 92)( 20, 91)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)( 26, 97)( 27, 99)( 28,110)( 29,109)( 30,111)( 31,113)( 32,112)( 33,114)( 34,116)( 35,115)( 36,117)( 37,128)( 38,127)( 39,129)( 40,131)( 41,130)( 42,132)( 43,134)( 44,133)( 45,135)( 46,119)( 47,118)( 48,120)( 49,122)( 50,121)( 51,123)( 52,125)( 53,124)( 54,126)( 55,137)( 56,136)( 57,138)( 58,140)( 59,139)( 60,141)( 61,143)( 62,142)( 63,144)( 64,155)( 65,154)( 66,156)( 67,158)( 68,157)( 69,159)( 70,161)( 71,160)( 72,162)( 73,146)( 74,145)( 75,147)( 76,149)( 77,148)( 78,150)( 79,152)( 80,151)( 81,153);;
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, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s4*s2*s3*s2*s3*s4*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(162)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)( 34, 58)( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)( 42, 72)( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)( 50, 80)( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159);
s1 := Sym(162)!(  1, 31)(  2, 32)(  3, 33)(  4, 28)(  5, 29)(  6, 30)(  7, 34)(  8, 35)(  9, 36)( 10, 40)( 11, 41)( 12, 42)( 13, 37)( 14, 38)( 15, 39)( 16, 43)( 17, 44)( 18, 45)( 19, 49)( 20, 50)( 21, 51)( 22, 46)( 23, 47)( 24, 48)( 25, 52)( 26, 53)( 27, 54)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)( 73, 76)( 74, 77)( 75, 78)( 82,112)( 83,113)( 84,114)( 85,109)( 86,110)( 87,111)( 88,115)( 89,116)( 90,117)( 91,121)( 92,122)( 93,123)( 94,118)( 95,119)( 96,120)( 97,124)( 98,125)( 99,126)(100,130)(101,131)(102,132)(103,127)(104,128)(105,129)(106,133)(107,134)(108,135)(136,139)(137,140)(138,141)(145,148)(146,149)(147,150)(154,157)(155,158)(156,159);
s2 := Sym(162)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)( 18, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)( 36, 63)( 37, 73)( 38, 74)( 39, 75)( 40, 76)( 41, 77)( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 64)( 47, 65)( 48, 66)( 49, 67)( 50, 68)( 51, 69)( 52, 70)( 53, 71)( 54, 72)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)(119,155)(120,156)(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153);
s3 := Sym(162)!(  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)(  9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)( 53, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)(116,126)(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)(139,148)(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)(161,162);
s4 := Sym(162)!(  1, 83)(  2, 82)(  3, 84)(  4, 86)(  5, 85)(  6, 87)(  7, 89)(  8, 88)(  9, 90)( 10,101)( 11,100)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)( 18,108)( 19, 92)( 20, 91)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)( 26, 97)( 27, 99)( 28,110)( 29,109)( 30,111)( 31,113)( 32,112)( 33,114)( 34,116)( 35,115)( 36,117)( 37,128)( 38,127)( 39,129)( 40,131)( 41,130)( 42,132)( 43,134)( 44,133)( 45,135)( 46,119)( 47,118)( 48,120)( 49,122)( 50,121)( 51,123)( 52,125)( 53,124)( 54,126)( 55,137)( 56,136)( 57,138)( 58,140)( 59,139)( 60,141)( 61,143)( 62,142)( 63,144)( 64,155)( 65,154)( 66,156)( 67,158)( 68,157)( 69,159)( 70,161)( 71,160)( 72,162)( 73,146)( 74,145)( 75,147)( 76,149)( 77,148)( 78,150)( 79,152)( 80,151)( 81,153);
poly := sub<Sym(162)|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, s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s4*s2*s3*s2*s3*s4*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 >;

References