Polytope of Type {6,6,26} with group SmallGroup(1872,1061)

Overview

Group: SmallGroup(1872,1061)
Rank: 4
Schläfli Type: {6,6,26}
Vertices, edges, …: 6, 18, 78, 26
Order of s₀s₁s₂s₃: 78
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{6,6|2},{6,26|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

{2,2,26}*208

12-fold

{3,2,13}*156

13-fold

{6,6,2}*144a

18-fold

{2,2,13}*104

39-fold

78-fold

117-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 := ( 14, 27)( 15, 28)( 16, 29)( 17, 30)( 18, 31)( 19, 32)( 20, 33)( 21, 34)( 22, 35)( 23, 36)( 24, 37)( 25, 38)( 26, 39)( 53, 66)( 54, 67)( 55, 68)( 56, 69)( 57, 70)( 58, 71)( 59, 72)( 60, 73)( 61, 74)( 62, 75)( 63, 76)( 64, 77)( 65, 78)( 92,105)( 93,106)( 94,107)( 95,108)( 96,109)( 97,110)( 98,111)( 99,112)(100,113)(101,114)(102,115)(103,116)(104,117)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156)(170,183)(171,184)(172,185)(173,186)(174,187)(175,188)(176,189)(177,190)(178,191)(179,192)(180,193)(181,194)(182,195)(209,222)(210,223)(211,224)(212,225)(213,226)(214,227)(215,228)(216,229)(217,230)(218,231)(219,232)(220,233)(221,234);;
s1 := (  1, 14)(  2, 15)(  3, 16)(  4, 17)(  5, 18)(  6, 19)(  7, 20)(  8, 21)(  9, 22)( 10, 23)( 11, 24)( 12, 25)( 13, 26)( 40, 92)( 41, 93)( 42, 94)( 43, 95)( 44, 96)( 45, 97)( 46, 98)( 47, 99)( 48,100)( 49,101)( 50,102)( 51,103)( 52,104)( 53, 79)( 54, 80)( 55, 81)( 56, 82)( 57, 83)( 58, 84)( 59, 85)( 60, 86)( 61, 87)( 62, 88)( 63, 89)( 64, 90)( 65, 91)( 66,105)( 67,106)( 68,107)( 69,108)( 70,109)( 71,110)( 72,111)( 73,112)( 74,113)( 75,114)( 76,115)( 77,116)( 78,117)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141)(129,142)(130,143)(157,209)(158,210)(159,211)(160,212)(161,213)(162,214)(163,215)(164,216)(165,217)(166,218)(167,219)(168,220)(169,221)(170,196)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234);;
s2 := (  1, 40)(  2, 52)(  3, 51)(  4, 50)(  5, 49)(  6, 48)(  7, 47)(  8, 46)(  9, 45)( 10, 44)( 11, 43)( 12, 42)( 13, 41)( 14, 53)( 15, 65)( 16, 64)( 17, 63)( 18, 62)( 19, 61)( 20, 60)( 21, 59)( 22, 58)( 23, 57)( 24, 56)( 25, 55)( 26, 54)( 27, 66)( 28, 78)( 29, 77)( 30, 76)( 31, 75)( 32, 74)( 33, 73)( 34, 72)( 35, 71)( 36, 70)( 37, 69)( 38, 68)( 39, 67)( 80, 91)( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 93,104)( 94,103)( 95,102)( 96,101)( 97,100)( 98, 99)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(118,157)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,170)(132,182)(133,181)(134,180)(135,179)(136,178)(137,177)(138,176)(139,175)(140,174)(141,173)(142,172)(143,171)(144,183)(145,195)(146,194)(147,193)(148,192)(149,191)(150,190)(151,189)(152,188)(153,187)(154,186)(155,185)(156,184)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(210,221)(211,220)(212,219)(213,218)(214,217)(215,216)(223,234)(224,233)(225,232)(226,231)(227,230)(228,229);;
s3 := (  1,119)(  2,118)(  3,130)(  4,129)(  5,128)(  6,127)(  7,126)(  8,125)(  9,124)( 10,123)( 11,122)( 12,121)( 13,120)( 14,132)( 15,131)( 16,143)( 17,142)( 18,141)( 19,140)( 20,139)( 21,138)( 22,137)( 23,136)( 24,135)( 25,134)( 26,133)( 27,145)( 28,144)( 29,156)( 30,155)( 31,154)( 32,153)( 33,152)( 34,151)( 35,150)( 36,149)( 37,148)( 38,147)( 39,146)( 40,158)( 41,157)( 42,169)( 43,168)( 44,167)( 45,166)( 46,165)( 47,164)( 48,163)( 49,162)( 50,161)( 51,160)( 52,159)( 53,171)( 54,170)( 55,182)( 56,181)( 57,180)( 58,179)( 59,178)( 60,177)( 61,176)( 62,175)( 63,174)( 64,173)( 65,172)( 66,184)( 67,183)( 68,195)( 69,194)( 70,193)( 71,192)( 72,191)( 73,190)( 74,189)( 75,188)( 76,187)( 77,186)( 78,185)( 79,197)( 80,196)( 81,208)( 82,207)( 83,206)( 84,205)( 85,204)( 86,203)( 87,202)( 88,201)( 89,200)( 90,199)( 91,198)( 92,210)( 93,209)( 94,221)( 95,220)( 96,219)( 97,218)( 98,217)( 99,216)(100,215)(101,214)(102,213)(103,212)(104,211)(105,223)(106,222)(107,234)(108,233)(109,232)(110,231)(111,230)(112,229)(113,228)(114,227)(115,226)(116,225)(117,224);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(234)!( 14, 27)( 15, 28)( 16, 29)( 17, 30)( 18, 31)( 19, 32)( 20, 33)( 21, 34)( 22, 35)( 23, 36)( 24, 37)( 25, 38)( 26, 39)( 53, 66)( 54, 67)( 55, 68)( 56, 69)( 57, 70)( 58, 71)( 59, 72)( 60, 73)( 61, 74)( 62, 75)( 63, 76)( 64, 77)( 65, 78)( 92,105)( 93,106)( 94,107)( 95,108)( 96,109)( 97,110)( 98,111)( 99,112)(100,113)(101,114)(102,115)(103,116)(104,117)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156)(170,183)(171,184)(172,185)(173,186)(174,187)(175,188)(176,189)(177,190)(178,191)(179,192)(180,193)(181,194)(182,195)(209,222)(210,223)(211,224)(212,225)(213,226)(214,227)(215,228)(216,229)(217,230)(218,231)(219,232)(220,233)(221,234);
s1 := Sym(234)!(  1, 14)(  2, 15)(  3, 16)(  4, 17)(  5, 18)(  6, 19)(  7, 20)(  8, 21)(  9, 22)( 10, 23)( 11, 24)( 12, 25)( 13, 26)( 40, 92)( 41, 93)( 42, 94)( 43, 95)( 44, 96)( 45, 97)( 46, 98)( 47, 99)( 48,100)( 49,101)( 50,102)( 51,103)( 52,104)( 53, 79)( 54, 80)( 55, 81)( 56, 82)( 57, 83)( 58, 84)( 59, 85)( 60, 86)( 61, 87)( 62, 88)( 63, 89)( 64, 90)( 65, 91)( 66,105)( 67,106)( 68,107)( 69,108)( 70,109)( 71,110)( 72,111)( 73,112)( 74,113)( 75,114)( 76,115)( 77,116)( 78,117)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141)(129,142)(130,143)(157,209)(158,210)(159,211)(160,212)(161,213)(162,214)(163,215)(164,216)(165,217)(166,218)(167,219)(168,220)(169,221)(170,196)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234);
s2 := Sym(234)!(  1, 40)(  2, 52)(  3, 51)(  4, 50)(  5, 49)(  6, 48)(  7, 47)(  8, 46)(  9, 45)( 10, 44)( 11, 43)( 12, 42)( 13, 41)( 14, 53)( 15, 65)( 16, 64)( 17, 63)( 18, 62)( 19, 61)( 20, 60)( 21, 59)( 22, 58)( 23, 57)( 24, 56)( 25, 55)( 26, 54)( 27, 66)( 28, 78)( 29, 77)( 30, 76)( 31, 75)( 32, 74)( 33, 73)( 34, 72)( 35, 71)( 36, 70)( 37, 69)( 38, 68)( 39, 67)( 80, 91)( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 93,104)( 94,103)( 95,102)( 96,101)( 97,100)( 98, 99)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(118,157)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,170)(132,182)(133,181)(134,180)(135,179)(136,178)(137,177)(138,176)(139,175)(140,174)(141,173)(142,172)(143,171)(144,183)(145,195)(146,194)(147,193)(148,192)(149,191)(150,190)(151,189)(152,188)(153,187)(154,186)(155,185)(156,184)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(210,221)(211,220)(212,219)(213,218)(214,217)(215,216)(223,234)(224,233)(225,232)(226,231)(227,230)(228,229);
s3 := Sym(234)!(  1,119)(  2,118)(  3,130)(  4,129)(  5,128)(  6,127)(  7,126)(  8,125)(  9,124)( 10,123)( 11,122)( 12,121)( 13,120)( 14,132)( 15,131)( 16,143)( 17,142)( 18,141)( 19,140)( 20,139)( 21,138)( 22,137)( 23,136)( 24,135)( 25,134)( 26,133)( 27,145)( 28,144)( 29,156)( 30,155)( 31,154)( 32,153)( 33,152)( 34,151)( 35,150)( 36,149)( 37,148)( 38,147)( 39,146)( 40,158)( 41,157)( 42,169)( 43,168)( 44,167)( 45,166)( 46,165)( 47,164)( 48,163)( 49,162)( 50,161)( 51,160)( 52,159)( 53,171)( 54,170)( 55,182)( 56,181)( 57,180)( 58,179)( 59,178)( 60,177)( 61,176)( 62,175)( 63,174)( 64,173)( 65,172)( 66,184)( 67,183)( 68,195)( 69,194)( 70,193)( 71,192)( 72,191)( 73,190)( 74,189)( 75,188)( 76,187)( 77,186)( 78,185)( 79,197)( 80,196)( 81,208)( 82,207)( 83,206)( 84,205)( 85,204)( 86,203)( 87,202)( 88,201)( 89,200)( 90,199)( 91,198)( 92,210)( 93,209)( 94,221)( 95,220)( 96,219)( 97,218)( 98,217)( 99,216)(100,215)(101,214)(102,213)(103,212)(104,211)(105,223)(106,222)(107,234)(108,233)(109,232)(110,231)(111,230)(112,229)(113,228)(114,227)(115,226)(116,225)(117,224);
poly := sub<Sym(234)|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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References