Polytope of Type {6,6,6,4} with group SmallGroup(1728,47409)

Overview

Group: SmallGroup(1728,47409)
Rank: 5
Schläfli Type: {6,6,6,4}
Vertices, edges, …: 6, 18, 18, 12, 4
Order of s₀s₁s₂s₃s₄: 12
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

{6,6,6,2}*864e

3-fold

6-fold

9-fold

12-fold

{2,3,6,2}*144

18-fold

27-fold

{2,2,2,4}*64

36-fold

54-fold

{2,2,2,2}*32

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

References