Polytope of Type {3,6,6} with group SmallGroup(648,299)

Overview

Group: SmallGroup(648,299)
Rank: 4
Schläfli Type: {3,6,6}
Vertices, edges, …: 3, 27, 54, 18
Order of s₀s₁s₂s₃: 18
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

{3,6,3}*324a

3-fold

{3,6,6}*216a

6-fold

{3,6,3}*108

9-fold

{3,2,6}*72

18-fold

{3,2,3}*36

27-fold

{3,2,2}*24

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s2)^2> of order 3

10 facets

6 of {3,2}*12
4 of {3,6}*36

3 vertex figures

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

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.