Polytope of Type {9,6,4,4} with group SmallGroup(1728,17334)

Overview

Group: SmallGroup(1728,17334)
Rank: 5
Schläfli Type: {9,6,4,4}
Vertices, edges, …: 9, 27, 12, 8, 4
Order of s₀s₁s₂s₃s₄: 36
Order of s₀s₁s₂s₃s₄s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

{9,6,2,2}*432

6-fold

9-fold

{3,2,4,4}*192

12-fold

18-fold

36-fold

{3,2,2,2}*48

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 21)( 11, 20)( 12, 19)( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)( 43, 51)( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)( 65, 74)( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)(114,116)(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)(147,154)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(164,165)(166,169)(167,171)(168,170)(172,183)(173,182)(174,181)(175,189)(176,188)(177,187)(178,186)(179,185)(180,184)(191,192)(193,196)(194,198)(195,197)(199,210)(200,209)(201,208)(202,216)(203,215)(204,214)(205,213)(206,212)(207,211);;
s1 := (  1, 13)(  2, 15)(  3, 14)(  4, 10)(  5, 12)(  6, 11)(  7, 16)(  8, 18)(  9, 17)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 37)( 32, 39)( 33, 38)( 34, 43)( 35, 45)( 36, 44)( 46, 51)( 47, 50)( 48, 49)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 61, 70)( 62, 72)( 63, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82, 94)( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 97)( 89, 99)( 90, 98)(100,105)(101,104)(102,103)(106,108)(109,121)(110,123)(111,122)(112,118)(113,120)(114,119)(115,124)(116,126)(117,125)(127,132)(128,131)(129,130)(133,135)(136,148)(137,150)(138,149)(139,145)(140,147)(141,146)(142,151)(143,153)(144,152)(154,159)(155,158)(156,157)(160,162)(163,175)(164,177)(165,176)(166,172)(167,174)(168,173)(169,178)(170,180)(171,179)(181,186)(182,185)(183,184)(187,189)(190,202)(191,204)(192,203)(193,199)(194,201)(195,200)(196,205)(197,207)(198,206)(208,213)(209,212)(210,211)(214,216);;
s2 := (  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)(109,136)(110,137)(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)(188,212)(189,213);;
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)( 55,163)( 56,164)( 57,165)( 58,166)( 59,167)( 60,168)( 61,169)( 62,170)( 63,171)( 64,172)( 65,173)( 66,174)( 67,175)( 68,176)( 69,177)( 70,178)( 71,179)( 72,180)( 73,181)( 74,182)( 75,183)( 76,184)( 77,185)( 78,186)( 79,187)( 80,188)( 81,189)( 82,190)( 83,191)( 84,192)( 85,193)( 86,194)( 87,195)( 88,196)( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)( 94,202)( 95,203)( 96,204)( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)(106,214)(107,215)(108,216);;
s4 := (109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(216)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 21)( 11, 20)( 12, 19)( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)( 43, 51)( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)( 65, 74)( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)(114,116)(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)(147,154)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(164,165)(166,169)(167,171)(168,170)(172,183)(173,182)(174,181)(175,189)(176,188)(177,187)(178,186)(179,185)(180,184)(191,192)(193,196)(194,198)(195,197)(199,210)(200,209)(201,208)(202,216)(203,215)(204,214)(205,213)(206,212)(207,211);
s1 := Sym(216)!(  1, 13)(  2, 15)(  3, 14)(  4, 10)(  5, 12)(  6, 11)(  7, 16)(  8, 18)(  9, 17)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 37)( 32, 39)( 33, 38)( 34, 43)( 35, 45)( 36, 44)( 46, 51)( 47, 50)( 48, 49)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 61, 70)( 62, 72)( 63, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82, 94)( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 97)( 89, 99)( 90, 98)(100,105)(101,104)(102,103)(106,108)(109,121)(110,123)(111,122)(112,118)(113,120)(114,119)(115,124)(116,126)(117,125)(127,132)(128,131)(129,130)(133,135)(136,148)(137,150)(138,149)(139,145)(140,147)(141,146)(142,151)(143,153)(144,152)(154,159)(155,158)(156,157)(160,162)(163,175)(164,177)(165,176)(166,172)(167,174)(168,173)(169,178)(170,180)(171,179)(181,186)(182,185)(183,184)(187,189)(190,202)(191,204)(192,203)(193,199)(194,201)(195,200)(196,205)(197,207)(198,206)(208,213)(209,212)(210,211)(214,216);
s2 := 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)(109,136)(110,137)(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)(188,212)(189,213);
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)( 55,163)( 56,164)( 57,165)( 58,166)( 59,167)( 60,168)( 61,169)( 62,170)( 63,171)( 64,172)( 65,173)( 66,174)( 67,175)( 68,176)( 69,177)( 70,178)( 71,179)( 72,180)( 73,181)( 74,182)( 75,183)( 76,184)( 77,185)( 78,186)( 79,187)( 80,188)( 81,189)( 82,190)( 83,191)( 84,192)( 85,193)( 86,194)( 87,195)( 88,196)( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)( 94,202)( 95,203)( 96,204)( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)(106,214)(107,215)(108,216);
s4 := Sym(216)!(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);
poly := sub<Sym(216)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References