Polytope of Type {3,6,6,6} with group SmallGroup(1728,47874)

Overview

Group: SmallGroup(1728,47874)
Rank: 5
Schläfli Type: {3,6,6,6}
Vertices, edges, …: 4, 12, 24, 18, 6
Order of s₀s₁s₂s₃s₄: 12
Order of s₀s₁s₂s₃s₄s₃s₂s₁: 2
Also known as: {{3,6}₄,{6,6|2},{6,6|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

{3,6,2,2}*192

12-fold

{3,3,2,3}*144

18-fold

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

Permutation Representation (Magma)

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

References