Polytope of Type {4,36,6} with group SmallGroup(1728,14461)

Overview

Group: SmallGroup(1728,14461)
Rank: 4
Schläfli Type: {4,36,6}
Vertices, edges, …: 4, 72, 108, 6
Order of s₀s₁s₂s₃: 36
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{4,36|2},{36,6|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

{2,18,6}*432a

6-fold

9-fold

12-fold

18-fold

24-fold

{2,9,2}*72

27-fold

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

Permutation Representation (Magma)

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

References