Polytope of Type {6,6,18} with group SmallGroup(1296,2984)

Overview

Group: SmallGroup(1296,2984)
Rank: 4
Schläfli Type: {6,6,18}
Vertices, edges, …: 6, 18, 54, 18
Order of s₀s₁s₂s₃: 18
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

9-fold

18-fold

27-fold

54-fold

81-fold

{2,2,2}*16

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References