Polytope of Type {6,18,8} with group SmallGroup(1728,15957)

Overview

Group: SmallGroup(1728,15957)
Rank: 4
Schläfli Type: {6,18,8}
Vertices, edges, …: 6, 54, 72, 8
Order of s₀s₁s₂s₃: 72
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{6,18|2},{18,8|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{6,18,4}*864a

3-fold

4-fold

{6,18,2}*432a

6-fold

9-fold

12-fold

18-fold

24-fold

{2,9,2}*72

27-fold

{2,2,8}*64

36-fold

54-fold

{2,2,4}*32

72-fold

108-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 := (  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)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 24)( 11, 23)( 12, 22)( 13, 21)( 14, 20)( 15, 19)( 16, 27)( 17, 26)( 18, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)( 43, 54)( 44, 53)( 45, 52)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 78)( 65, 77)( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 81)( 71, 80)( 72, 79)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91,105)( 92,104)( 93,103)( 94,102)( 95,101)( 96,100)( 97,108)( 98,107)( 99,106)(109,112)(110,114)(111,113)(116,117)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,135)(125,134)(126,133)(136,139)(137,141)(138,140)(143,144)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,162)(152,161)(153,160)(163,166)(164,168)(165,167)(170,171)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,189)(179,188)(180,187)(190,193)(191,195)(192,194)(197,198)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,216)(206,215)(207,214);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)(  9, 17)( 19, 21)( 22, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 46, 48)( 49, 51)( 52, 54)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 96)( 60, 95)( 61, 97)( 62, 99)( 63, 98)( 64, 82)( 65, 84)( 66, 83)( 67, 85)( 68, 87)( 69, 86)( 70, 88)( 71, 90)( 72, 89)( 73,102)( 74,101)( 75,100)( 76,105)( 77,104)( 78,103)( 79,108)( 80,107)( 81,106)(109,172)(110,174)(111,173)(112,175)(113,177)(114,176)(115,178)(116,180)(117,179)(118,163)(119,165)(120,164)(121,166)(122,168)(123,167)(124,169)(125,171)(126,170)(127,183)(128,182)(129,181)(130,186)(131,185)(132,184)(133,189)(134,188)(135,187)(136,199)(137,201)(138,200)(139,202)(140,204)(141,203)(142,205)(143,207)(144,206)(145,190)(146,192)(147,191)(148,193)(149,195)(150,194)(151,196)(152,198)(153,197)(154,210)(155,209)(156,208)(157,213)(158,212)(159,211)(160,216)(161,215)(162,214);;
s3 := (  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)(  8,116)(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189);;
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*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(216)!(  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)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);
s1 := Sym(216)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 24)( 11, 23)( 12, 22)( 13, 21)( 14, 20)( 15, 19)( 16, 27)( 17, 26)( 18, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)( 43, 54)( 44, 53)( 45, 52)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 78)( 65, 77)( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 81)( 71, 80)( 72, 79)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91,105)( 92,104)( 93,103)( 94,102)( 95,101)( 96,100)( 97,108)( 98,107)( 99,106)(109,112)(110,114)(111,113)(116,117)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,135)(125,134)(126,133)(136,139)(137,141)(138,140)(143,144)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,162)(152,161)(153,160)(163,166)(164,168)(165,167)(170,171)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,189)(179,188)(180,187)(190,193)(191,195)(192,194)(197,198)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,216)(206,215)(207,214);
s2 := Sym(216)!(  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)(  9, 17)( 19, 21)( 22, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 46, 48)( 49, 51)( 52, 54)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 96)( 60, 95)( 61, 97)( 62, 99)( 63, 98)( 64, 82)( 65, 84)( 66, 83)( 67, 85)( 68, 87)( 69, 86)( 70, 88)( 71, 90)( 72, 89)( 73,102)( 74,101)( 75,100)( 76,105)( 77,104)( 78,103)( 79,108)( 80,107)( 81,106)(109,172)(110,174)(111,173)(112,175)(113,177)(114,176)(115,178)(116,180)(117,179)(118,163)(119,165)(120,164)(121,166)(122,168)(123,167)(124,169)(125,171)(126,170)(127,183)(128,182)(129,181)(130,186)(131,185)(132,184)(133,189)(134,188)(135,187)(136,199)(137,201)(138,200)(139,202)(140,204)(141,203)(142,205)(143,207)(144,206)(145,190)(146,192)(147,191)(148,193)(149,195)(150,194)(151,196)(152,198)(153,197)(154,210)(155,209)(156,208)(157,213)(158,212)(159,211)(160,216)(161,215)(162,214);
s3 := Sym(216)!(  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)(  8,116)(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189);
poly := sub<Sym(216)|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*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References