Polytope of Type {6,6,6} with group SmallGroup(1296,2985)

Overview

Group: SmallGroup(1296,2985)
Rank: 4
Schläfli Type: {6,6,6}
Vertices, edges, …: 6, 54, 54, 18
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{6,6,6}*648a

3-fold

6-fold

{2,6,6}*216

9-fold

18-fold

{6,3,2}*72

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

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s2*s1*s2*s3*s2*s1*s3*s2> of order 3

6 facets

6 of {6,6}*72b

6 vertex figures

6 of 3-fold non-regular quotient of {6,6}*216b

Representations

Permutation Representation (GAP)

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

References

None.

to this polytope.