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Polytope of Type {6,6,26}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,26}*1872b
if this polytope has a name.
Group : SmallGroup(1872,1061)
Rank : 4
Schlafli Type : {6,6,26}
Number of vertices, edges, etc : 6, 18, 78, 26
Order of s0s1s2s3 : 78
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,26}*624
   9-fold quotients : {2,2,26}*208
   13-fold quotients : {6,6,2}*144b
   18-fold quotients : {2,2,13}*104
   26-fold quotients : {6,3,2}*72
   39-fold quotients : {2,6,2}*48
   78-fold quotients : {2,3,2}*24
   117-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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, 66)( 15, 78)( 16, 77)
( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 71)( 23, 70)( 24, 69)
( 25, 68)( 26, 67)( 27, 53)( 28, 65)( 29, 64)( 30, 63)( 31, 62)( 32, 61)
( 33, 60)( 34, 59)( 35, 58)( 36, 57)( 37, 56)( 38, 55)( 39, 54)( 80, 91)
( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)( 94,116)
( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)(102,108)
(103,107)(104,106)(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)(131,183)
(132,195)(133,194)(134,193)(135,192)(136,191)(137,190)(138,189)(139,188)
(140,187)(141,186)(142,185)(143,184)(144,170)(145,182)(146,181)(147,180)
(148,179)(149,178)(150,177)(151,176)(152,175)(153,174)(154,173)(155,172)
(156,171)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,222)
(210,234)(211,233)(212,232)(213,231)(214,230)(215,229)(216,228)(217,227)
(218,226)(219,225)(220,224)(221,223);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 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, 66)( 15, 78)
( 16, 77)( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 71)( 23, 70)
( 24, 69)( 25, 68)( 26, 67)( 27, 53)( 28, 65)( 29, 64)( 30, 63)( 31, 62)
( 32, 61)( 33, 60)( 34, 59)( 35, 58)( 36, 57)( 37, 56)( 38, 55)( 39, 54)
( 80, 91)( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)
( 94,116)( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)
(102,108)(103,107)(104,106)(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)
(131,183)(132,195)(133,194)(134,193)(135,192)(136,191)(137,190)(138,189)
(139,188)(140,187)(141,186)(142,185)(143,184)(144,170)(145,182)(146,181)
(147,180)(148,179)(149,178)(150,177)(151,176)(152,175)(153,174)(154,173)
(155,172)(156,171)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)
(209,222)(210,234)(211,233)(212,232)(213,231)(214,230)(215,229)(216,228)
(217,227)(218,226)(219,225)(220,224)(221,223);
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
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 : None.
to this polytope