Polytope of Type {6,4,3,6} with group SmallGroup(1728,47874)

Overview

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

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

4-fold

{6,2,3,6}*432

6-fold

{2,4,3,6}*288

8-fold

{3,2,3,6}*216

9-fold

{2,4,3,2}*192

12-fold

18-fold

{2,4,3,2}*96

24-fold

{3,2,3,2}*72

36-fold

{2,2,3,2}*48

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s2)^2> of order 2

6 facets

6 of 2-fold non-regular quotient of {6,4,3}*288

6 vertex figures

6 of 2-fold non-regular quotient of {4,3,6}*288

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.