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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}*1872c
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) :
   2-fold quotients : {3,6,26}*936
   3-fold quotients : {6,2,26}*624
   6-fold quotients : {3,2,26}*312, {6,2,13}*312
   9-fold quotients : {2,2,26}*208
   12-fold quotients : {3,2,13}*156
   13-fold quotients : {6,6,2}*144c
   18-fold quotients : {2,2,13}*104
   26-fold quotients : {3,6,2}*72
   39-fold quotients : {6,2,2}*48
   78-fold quotients : {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)( 40, 79)( 41, 80)( 42, 81)
( 43, 82)( 44, 83)( 45, 84)( 46, 85)( 47, 86)( 48, 87)( 49, 88)( 50, 89)
( 51, 90)( 52, 91)( 53,105)( 54,106)( 55,107)( 56,108)( 57,109)( 58,110)
( 59,111)( 60,112)( 61,113)( 62,114)( 63,115)( 64,116)( 65,117)( 66, 92)
( 67, 93)( 68, 94)( 69, 95)( 70, 96)( 71, 97)( 72, 98)( 73, 99)( 74,100)
( 75,101)( 76,102)( 77,103)( 78,104)(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)(157,196)(158,197)(159,198)(160,199)(161,200)(162,201)(163,202)
(164,203)(165,204)(166,205)(167,206)(168,207)(169,208)(170,222)(171,223)
(172,224)(173,225)(174,226)(175,227)(176,228)(177,229)(178,230)(179,231)
(180,232)(181,233)(182,234)(183,209)(184,210)(185,211)(186,212)(187,213)
(188,214)(189,215)(190,216)(191,217)(192,218)(193,219)(194,220)(195,221);;
s1 := (  1,170)(  2,171)(  3,172)(  4,173)(  5,174)(  6,175)(  7,176)(  8,177)
(  9,178)( 10,179)( 11,180)( 12,181)( 13,182)( 14,157)( 15,158)( 16,159)
( 17,160)( 18,161)( 19,162)( 20,163)( 21,164)( 22,165)( 23,166)( 24,167)
( 25,168)( 26,169)( 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)( 40,131)
( 41,132)( 42,133)( 43,134)( 44,135)( 45,136)( 46,137)( 47,138)( 48,139)
( 49,140)( 50,141)( 51,142)( 52,143)( 53,118)( 54,119)( 55,120)( 56,121)
( 57,122)( 58,123)( 59,124)( 60,125)( 61,126)( 62,127)( 63,128)( 64,129)
( 65,130)( 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)( 79,209)( 80,210)
( 81,211)( 82,212)( 83,213)( 84,214)( 85,215)( 86,216)( 87,217)( 88,218)
( 89,219)( 90,220)( 91,221)( 92,196)( 93,197)( 94,198)( 95,199)( 96,200)
( 97,201)( 98,202)( 99,203)(100,204)(101,205)(102,206)(103,207)(104,208)
(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);;
s2 := (  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 14, 27)( 15, 39)
( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)( 23, 31)
( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)( 59, 73)
( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 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)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(131,144)
(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)
(140,148)(141,147)(142,146)(143,145)(158,169)(159,168)(160,167)(161,166)
(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)(175,191)
(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)(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,  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, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)( 46, 48)
( 53, 54)( 55, 65)( 56, 64)( 57, 63)( 58, 62)( 59, 61)( 66, 67)( 68, 78)
( 69, 77)( 70, 76)( 71, 75)( 72, 74)( 79, 80)( 81, 91)( 82, 90)( 83, 89)
( 84, 88)( 85, 87)( 92, 93)( 94,104)( 95,103)( 96,102)( 97,101)( 98,100)
(105,106)(107,117)(108,116)(109,115)(110,114)(111,113)(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)
(157,158)(159,169)(160,168)(161,167)(162,166)(163,165)(170,171)(172,182)
(173,181)(174,180)(175,179)(176,178)(183,184)(185,195)(186,194)(187,193)
(188,192)(189,191)(196,197)(198,208)(199,207)(200,206)(201,205)(202,204)
(209,210)(211,221)(212,220)(213,219)(214,218)(215,217)(222,223)(224,234)
(225,233)(226,232)(227,231)(228,230);;
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*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
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)( 40, 79)( 41, 80)
( 42, 81)( 43, 82)( 44, 83)( 45, 84)( 46, 85)( 47, 86)( 48, 87)( 49, 88)
( 50, 89)( 51, 90)( 52, 91)( 53,105)( 54,106)( 55,107)( 56,108)( 57,109)
( 58,110)( 59,111)( 60,112)( 61,113)( 62,114)( 63,115)( 64,116)( 65,117)
( 66, 92)( 67, 93)( 68, 94)( 69, 95)( 70, 96)( 71, 97)( 72, 98)( 73, 99)
( 74,100)( 75,101)( 76,102)( 77,103)( 78,104)(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)(157,196)(158,197)(159,198)(160,199)(161,200)(162,201)
(163,202)(164,203)(165,204)(166,205)(167,206)(168,207)(169,208)(170,222)
(171,223)(172,224)(173,225)(174,226)(175,227)(176,228)(177,229)(178,230)
(179,231)(180,232)(181,233)(182,234)(183,209)(184,210)(185,211)(186,212)
(187,213)(188,214)(189,215)(190,216)(191,217)(192,218)(193,219)(194,220)
(195,221);
s1 := Sym(234)!(  1,170)(  2,171)(  3,172)(  4,173)(  5,174)(  6,175)(  7,176)
(  8,177)(  9,178)( 10,179)( 11,180)( 12,181)( 13,182)( 14,157)( 15,158)
( 16,159)( 17,160)( 18,161)( 19,162)( 20,163)( 21,164)( 22,165)( 23,166)
( 24,167)( 25,168)( 26,169)( 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)
( 40,131)( 41,132)( 42,133)( 43,134)( 44,135)( 45,136)( 46,137)( 47,138)
( 48,139)( 49,140)( 50,141)( 51,142)( 52,143)( 53,118)( 54,119)( 55,120)
( 56,121)( 57,122)( 58,123)( 59,124)( 60,125)( 61,126)( 62,127)( 63,128)
( 64,129)( 65,130)( 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)( 79,209)
( 80,210)( 81,211)( 82,212)( 83,213)( 84,214)( 85,215)( 86,216)( 87,217)
( 88,218)( 89,219)( 90,220)( 91,221)( 92,196)( 93,197)( 94,198)( 95,199)
( 96,200)( 97,201)( 98,202)( 99,203)(100,204)(101,205)(102,206)(103,207)
(104,208)(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);
s2 := Sym(234)!(  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 14, 27)
( 15, 39)( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)
( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)
( 45, 48)( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)
( 59, 73)( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 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)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)
(131,144)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)
(139,149)(140,148)(141,147)(142,146)(143,145)(158,169)(159,168)(160,167)
(161,166)(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)
(175,191)(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)
(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,  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, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)
( 46, 48)( 53, 54)( 55, 65)( 56, 64)( 57, 63)( 58, 62)( 59, 61)( 66, 67)
( 68, 78)( 69, 77)( 70, 76)( 71, 75)( 72, 74)( 79, 80)( 81, 91)( 82, 90)
( 83, 89)( 84, 88)( 85, 87)( 92, 93)( 94,104)( 95,103)( 96,102)( 97,101)
( 98,100)(105,106)(107,117)(108,116)(109,115)(110,114)(111,113)(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)(157,158)(159,169)(160,168)(161,167)(162,166)(163,165)(170,171)
(172,182)(173,181)(174,180)(175,179)(176,178)(183,184)(185,195)(186,194)
(187,193)(188,192)(189,191)(196,197)(198,208)(199,207)(200,206)(201,205)
(202,204)(209,210)(211,221)(212,220)(213,219)(214,218)(215,217)(222,223)
(224,234)(225,233)(226,232)(227,231)(228,230);
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
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