Polytope of Type {6,12,12} with group SmallGroup(1728,37586)

Overview

Group: SmallGroup(1728,37586)
Rank: 4
Schläfli Type: {6,12,12}
Vertices, edges, …: 6, 36, 72, 12
Order of s₀s₁s₂s₃: 12
Order of s₀s₁s₂s₃s₂s₁: 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

{6,6,6}*432e

6-fold

9-fold

12-fold

18-fold

24-fold

{2,3,6}*72

27-fold

{2,4,4}*64

36-fold

54-fold

72-fold

108-fold

{2,2,2}*16

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma)

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

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2 >;

References