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

Overview

Group: SmallGroup(1728,47874)
Rank: 5
Schläfli Type: {3,6,6,4}
Vertices, edges, …: 3, 9, 36, 24, 8
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

2-fold

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

3-fold

{3,2,6,4}*576

4-fold

{3,6,6,2}*432b

6-fold

12-fold

24-fold

{3,2,3,2}*72

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

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s2*s3*s4*s3)^2> of order 2

4 facets

4 of {3,6,6}*216b

3 vertex figures

3 of 2-fold non-regular quotient of {6,6,4}*576a

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.