Polytope of Type {4,6,6,6} with group SmallGroup(1728,47874)

Overview

Group: SmallGroup(1728,47874)
Rank: 5
Schläfli Type: {4,6,6,6}
Vertices, edges, …: 4, 12, 18, 18, 6
Order of s₀s₁s₂s₃s₄: 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Non-Orientable
Flat

Quotients maximal quotients in bold

2-fold

{4,3,6,6}*864b

3-fold

6-fold

9-fold

{4,6,2,2}*192b

12-fold

{4,3,2,3}*144

18-fold

{4,3,2,2}*96

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References