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

Overview

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

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

{6,18,2}*432a

6-fold

9-fold

12-fold

18-fold

24-fold

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