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: 6
Also known as: if this polytope has a name.

Special Properties

Orientable
Flat

Quotients maximal quotients in bold

2-fold

{3,6,6}*648d

3-fold

{6,6,6}*432g

6-fold

{3,6,6}*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=<s2*s1*s3*s2*s1*s2*s3*s2> of order 3

10 facets

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

6 vertex figures

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

P/N, where N=<(s2*s3)^2> of order 3

6 facets

6 of {6,6}*72c

6 vertex figures

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

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.