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, …: 18, 54, 54, 6
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{6,6,3}*648d

3-fold

{6,6,6}*432c

6-fold

{6,6,3}*216b

9-fold

18-fold

27-fold

36-fold

{3,2,3}*36

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=<(s0*s1)^2> of order 3

6 facets

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

6 vertex figures

6 of {6,6}*72b

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3

6 facets

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

10 vertex figures

4 of {6,6}*72b
6 of {2,6}*24

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.