Polytope of Type {12,4,6,3} with group SmallGroup(1728,37587)

Overview

Group: SmallGroup(1728,37587)
Rank: 5
Schläfli Type: {12,4,6,3}
Vertices, edges, …: 12, 24, 12, 9, 3
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

3-fold

4-fold

{6,2,6,3}*432

6-fold

8-fold

{3,2,6,3}*216

9-fold

{4,4,2,3}*192

12-fold

18-fold

24-fold

{3,2,2,3}*72

36-fold

{2,2,2,3}*48

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References