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

Overview

Group: SmallGroup(1728,37586)
Rank: 4
Schläfli Type: {12,12,6}
Vertices, edges, …: 12, 72, 36, 6
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}*432g

6-fold

8-fold

{3,6,6}*216b

9-fold

12-fold

18-fold

24-fold

27-fold

{4,4,2}*64

36-fold

48-fold

{3,2,3}*36

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

Permutation Representation (Magma)

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

References