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Polytope of Type {26,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {26,6,6}*1872a
Also Known As : {{26,6|2},{6,6|2}}. if this polytope has another name.
Group : SmallGroup(1872,1061)
Rank : 4
Schlafli Type : {26,6,6}
Number of vertices, edges, etc : 26, 78, 18, 6
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 : {26,2,6}*624, {26,6,2}*624
   6-fold quotients : {13,2,6}*312, {26,2,3}*312
   9-fold quotients : {26,2,2}*208
   12-fold quotients : {13,2,3}*156
   13-fold quotients : {2,6,6}*144a
   18-fold quotients : {13,2,2}*104
   39-fold quotients : {2,2,6}*48, {2,6,2}*48
   78-fold quotients : {2,2,3}*24, {2,3,2}*24
   117-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 15, 26)( 16, 25)
( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 28, 39)( 29, 38)( 30, 37)( 31, 36)
( 32, 35)( 33, 34)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)( 46, 47)
( 54, 65)( 55, 64)( 56, 63)( 57, 62)( 58, 61)( 59, 60)( 67, 78)( 68, 77)
( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 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)(119,130)(120,129)
(121,128)(122,127)(123,126)(124,125)(132,143)(133,142)(134,141)(135,140)
(136,139)(137,138)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)
(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(171,182)(172,181)
(173,180)(174,179)(175,178)(176,177)(184,195)(185,194)(186,193)(187,192)
(188,191)(189,190)(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);;
s1 := (  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 28)( 15, 27)
( 16, 39)( 17, 38)( 18, 37)( 19, 36)( 20, 35)( 21, 34)( 22, 33)( 23, 32)
( 24, 31)( 25, 30)( 26, 29)( 40, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)
( 46, 48)( 53, 67)( 54, 66)( 55, 78)( 56, 77)( 57, 76)( 58, 75)( 59, 74)
( 60, 73)( 61, 72)( 62, 71)( 63, 70)( 64, 69)( 65, 68)( 79, 80)( 81, 91)
( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 92,106)( 93,105)( 94,117)( 95,116)
( 96,115)( 97,114)( 98,113)( 99,112)(100,111)(101,110)(102,109)(103,108)
(104,107)(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)(131,145)
(132,144)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)
(140,149)(141,148)(142,147)(143,146)(157,158)(159,169)(160,168)(161,167)
(162,166)(163,165)(170,184)(171,183)(172,195)(173,194)(174,193)(175,192)
(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(196,197)
(198,208)(199,207)(200,206)(201,205)(202,204)(209,223)(210,222)(211,234)
(212,233)(213,232)(214,231)(215,230)(216,229)(217,228)(218,227)(219,226)
(220,225)(221,224);;
s2 := (  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);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(234)!(  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 15, 26)
( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 28, 39)( 29, 38)( 30, 37)
( 31, 36)( 32, 35)( 33, 34)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 54, 65)( 55, 64)( 56, 63)( 57, 62)( 58, 61)( 59, 60)( 67, 78)
( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 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)(119,130)
(120,129)(121,128)(122,127)(123,126)(124,125)(132,143)(133,142)(134,141)
(135,140)(136,139)(137,138)(145,156)(146,155)(147,154)(148,153)(149,152)
(150,151)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(171,182)
(172,181)(173,180)(174,179)(175,178)(176,177)(184,195)(185,194)(186,193)
(187,192)(188,191)(189,190)(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);
s1 := Sym(234)!(  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 28)
( 15, 27)( 16, 39)( 17, 38)( 18, 37)( 19, 36)( 20, 35)( 21, 34)( 22, 33)
( 23, 32)( 24, 31)( 25, 30)( 26, 29)( 40, 41)( 42, 52)( 43, 51)( 44, 50)
( 45, 49)( 46, 48)( 53, 67)( 54, 66)( 55, 78)( 56, 77)( 57, 76)( 58, 75)
( 59, 74)( 60, 73)( 61, 72)( 62, 71)( 63, 70)( 64, 69)( 65, 68)( 79, 80)
( 81, 91)( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 92,106)( 93,105)( 94,117)
( 95,116)( 96,115)( 97,114)( 98,113)( 99,112)(100,111)(101,110)(102,109)
(103,108)(104,107)(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)
(131,145)(132,144)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)
(139,150)(140,149)(141,148)(142,147)(143,146)(157,158)(159,169)(160,168)
(161,167)(162,166)(163,165)(170,184)(171,183)(172,195)(173,194)(174,193)
(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)
(196,197)(198,208)(199,207)(200,206)(201,205)(202,204)(209,223)(210,222)
(211,234)(212,233)(213,232)(214,231)(215,230)(216,229)(217,228)(218,227)
(219,226)(220,225)(221,224);
s2 := 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);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope