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Polytope of Type {6,26,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,26,6}*1872
Also Known As : {{6,26|2},{26,6|2}}. if this polytope has another name.
Group : SmallGroup(1872,1061)
Rank : 4
Schlafli Type : {6,26,6}
Number of vertices, edges, etc : 6, 78, 78, 6
Order of s0s1s2s3 : 78
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
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 : {2,26,6}*624, {6,26,2}*624
   9-fold quotients : {2,26,2}*208
   13-fold quotients : {6,2,6}*144
   18-fold quotients : {2,13,2}*104
   26-fold quotients : {3,2,6}*72, {6,2,3}*72
   39-fold quotients : {2,2,6}*48, {6,2,2}*48
   52-fold quotients : {3,2,3}*36
   78-fold quotients : {2,2,3}*24, {3,2,2}*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, 26)(  3, 25)(  4, 24)(  5, 23)(  6, 22)(  7, 21)(  8, 20)
(  9, 19)( 10, 18)( 11, 17)( 12, 16)( 13, 15)( 28, 39)( 29, 38)( 30, 37)
( 31, 36)( 32, 35)( 33, 34)( 40, 53)( 41, 65)( 42, 64)( 43, 63)( 44, 62)
( 45, 61)( 46, 60)( 47, 59)( 48, 58)( 49, 57)( 50, 56)( 51, 55)( 52, 54)
( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 79, 92)( 80,104)
( 81,103)( 82,102)( 83,101)( 84,100)( 85, 99)( 86, 98)( 87, 97)( 88, 96)
( 89, 95)( 90, 94)( 91, 93)(106,117)(107,116)(108,115)(109,114)(110,113)
(111,112)(118,131)(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)
(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(145,156)(146,155)
(147,154)(148,153)(149,152)(150,151)(157,170)(158,182)(159,181)(160,180)
(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)(168,172)
(169,171)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(196,209)
(197,221)(198,220)(199,219)(200,218)(201,217)(202,216)(203,215)(204,214)
(205,213)(206,212)(207,211)(208,210)(223,234)(224,233)(225,232)(226,231)
(227,230)(228,229);;
s2 := (  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 15)( 16, 26)
( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 28)( 29, 39)( 30, 38)( 31, 37)
( 32, 36)( 33, 35)( 40, 80)( 41, 79)( 42, 91)( 43, 90)( 44, 89)( 45, 88)
( 46, 87)( 47, 86)( 48, 85)( 49, 84)( 50, 83)( 51, 82)( 52, 81)( 53, 93)
( 54, 92)( 55,104)( 56,103)( 57,102)( 58,101)( 59,100)( 60, 99)( 61, 98)
( 62, 97)( 63, 96)( 64, 95)( 65, 94)( 66,106)( 67,105)( 68,117)( 69,116)
( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)( 76,109)( 77,108)
( 78,107)(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)(131,132)
(133,143)(134,142)(135,141)(136,140)(137,139)(144,145)(146,156)(147,155)
(148,154)(149,153)(150,152)(157,197)(158,196)(159,208)(160,207)(161,206)
(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)
(170,210)(171,209)(172,221)(173,220)(174,219)(175,218)(176,217)(177,216)
(178,215)(179,214)(180,213)(181,212)(182,211)(183,223)(184,222)(185,234)
(186,233)(187,232)(188,231)(189,230)(190,229)(191,228)(192,227)(193,226)
(194,225)(195,224);;
s3 := (  1,157)(  2,158)(  3,159)(  4,160)(  5,161)(  6,162)(  7,163)(  8,164)
(  9,165)( 10,166)( 11,167)( 12,168)( 13,169)( 14,170)( 15,171)( 16,172)
( 17,173)( 18,174)( 19,175)( 20,176)( 21,177)( 22,178)( 23,179)( 24,180)
( 25,181)( 26,182)( 27,183)( 28,184)( 29,185)( 30,186)( 31,187)( 32,188)
( 33,189)( 34,190)( 35,191)( 36,192)( 37,193)( 38,194)( 39,195)( 40,118)
( 41,119)( 42,120)( 43,121)( 44,122)( 45,123)( 46,124)( 47,125)( 48,126)
( 49,127)( 50,128)( 51,129)( 52,130)( 53,131)( 54,132)( 55,133)( 56,134)
( 57,135)( 58,136)( 59,137)( 60,138)( 61,139)( 62,140)( 63,141)( 64,142)
( 65,143)( 66,144)( 67,145)( 68,146)( 69,147)( 70,148)( 71,149)( 72,150)
( 73,151)( 74,152)( 75,153)( 76,154)( 77,155)( 78,156)( 79,196)( 80,197)
( 81,198)( 82,199)( 83,200)( 84,201)( 85,202)( 86,203)( 87,204)( 88,205)
( 89,206)( 90,207)( 91,208)( 92,209)( 93,210)( 94,211)( 95,212)( 96,213)
( 97,214)( 98,215)( 99,216)(100,217)(101,218)(102,219)(103,220)(104,221)
(105,222)(106,223)(107,224)(108,225)(109,226)(110,227)(111,228)(112,229)
(113,230)(114,231)(115,232)(116,233)(117,234);;
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, 
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 ];;
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, 26)(  3, 25)(  4, 24)(  5, 23)(  6, 22)(  7, 21)
(  8, 20)(  9, 19)( 10, 18)( 11, 17)( 12, 16)( 13, 15)( 28, 39)( 29, 38)
( 30, 37)( 31, 36)( 32, 35)( 33, 34)( 40, 53)( 41, 65)( 42, 64)( 43, 63)
( 44, 62)( 45, 61)( 46, 60)( 47, 59)( 48, 58)( 49, 57)( 50, 56)( 51, 55)
( 52, 54)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 79, 92)
( 80,104)( 81,103)( 82,102)( 83,101)( 84,100)( 85, 99)( 86, 98)( 87, 97)
( 88, 96)( 89, 95)( 90, 94)( 91, 93)(106,117)(107,116)(108,115)(109,114)
(110,113)(111,112)(118,131)(119,143)(120,142)(121,141)(122,140)(123,139)
(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(145,156)
(146,155)(147,154)(148,153)(149,152)(150,151)(157,170)(158,182)(159,181)
(160,180)(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)
(168,172)(169,171)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)
(196,209)(197,221)(198,220)(199,219)(200,218)(201,217)(202,216)(203,215)
(204,214)(205,213)(206,212)(207,211)(208,210)(223,234)(224,233)(225,232)
(226,231)(227,230)(228,229);
s2 := Sym(234)!(  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 15)
( 16, 26)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 28)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 40, 80)( 41, 79)( 42, 91)( 43, 90)( 44, 89)
( 45, 88)( 46, 87)( 47, 86)( 48, 85)( 49, 84)( 50, 83)( 51, 82)( 52, 81)
( 53, 93)( 54, 92)( 55,104)( 56,103)( 57,102)( 58,101)( 59,100)( 60, 99)
( 61, 98)( 62, 97)( 63, 96)( 64, 95)( 65, 94)( 66,106)( 67,105)( 68,117)
( 69,116)( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)( 76,109)
( 77,108)( 78,107)(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)
(131,132)(133,143)(134,142)(135,141)(136,140)(137,139)(144,145)(146,156)
(147,155)(148,154)(149,153)(150,152)(157,197)(158,196)(159,208)(160,207)
(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)
(169,198)(170,210)(171,209)(172,221)(173,220)(174,219)(175,218)(176,217)
(177,216)(178,215)(179,214)(180,213)(181,212)(182,211)(183,223)(184,222)
(185,234)(186,233)(187,232)(188,231)(189,230)(190,229)(191,228)(192,227)
(193,226)(194,225)(195,224);
s3 := Sym(234)!(  1,157)(  2,158)(  3,159)(  4,160)(  5,161)(  6,162)(  7,163)
(  8,164)(  9,165)( 10,166)( 11,167)( 12,168)( 13,169)( 14,170)( 15,171)
( 16,172)( 17,173)( 18,174)( 19,175)( 20,176)( 21,177)( 22,178)( 23,179)
( 24,180)( 25,181)( 26,182)( 27,183)( 28,184)( 29,185)( 30,186)( 31,187)
( 32,188)( 33,189)( 34,190)( 35,191)( 36,192)( 37,193)( 38,194)( 39,195)
( 40,118)( 41,119)( 42,120)( 43,121)( 44,122)( 45,123)( 46,124)( 47,125)
( 48,126)( 49,127)( 50,128)( 51,129)( 52,130)( 53,131)( 54,132)( 55,133)
( 56,134)( 57,135)( 58,136)( 59,137)( 60,138)( 61,139)( 62,140)( 63,141)
( 64,142)( 65,143)( 66,144)( 67,145)( 68,146)( 69,147)( 70,148)( 71,149)
( 72,150)( 73,151)( 74,152)( 75,153)( 76,154)( 77,155)( 78,156)( 79,196)
( 80,197)( 81,198)( 82,199)( 83,200)( 84,201)( 85,202)( 86,203)( 87,204)
( 88,205)( 89,206)( 90,207)( 91,208)( 92,209)( 93,210)( 94,211)( 95,212)
( 96,213)( 97,214)( 98,215)( 99,216)(100,217)(101,218)(102,219)(103,220)
(104,221)(105,222)(106,223)(107,224)(108,225)(109,226)(110,227)(111,228)
(112,229)(113,230)(114,231)(115,232)(116,233)(117,234);
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, 
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 >; 
 
References : None.
to this polytope