Polytope of Type {26,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {26,6,6}*1872c
if this polytope has a 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 : {26,6,2}*624
   9-fold quotients : {26,2,2}*208
   13-fold quotients : {2,6,6}*144c
   18-fold quotients : {13,2,2}*104
   26-fold quotients : {2,3,6}*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 := (  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, 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)
( 53,106)( 54,105)( 55,117)( 56,116)( 57,115)( 58,114)( 59,113)( 60,112)
( 61,111)( 62,110)( 63,109)( 64,108)( 65,107)( 66, 93)( 67, 92)( 68,104)
( 69,103)( 70,102)( 71,101)( 72,100)( 73, 99)( 74, 98)( 75, 97)( 76, 96)
( 77, 95)( 78, 94)(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)(157,197)(158,196)(159,208)
(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)
(168,199)(169,198)(170,223)(171,222)(172,234)(173,233)(174,232)(175,231)
(176,230)(177,229)(178,228)(179,227)(180,226)(181,225)(182,224)(183,210)
(184,209)(185,221)(186,220)(187,219)(188,218)(189,217)(190,216)(191,215)
(192,214)(193,213)(194,212)(195,211);;
s2 := (  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);;
s3 := ( 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);;
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
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, 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)( 53,106)( 54,105)( 55,117)( 56,116)( 57,115)( 58,114)( 59,113)
( 60,112)( 61,111)( 62,110)( 63,109)( 64,108)( 65,107)( 66, 93)( 67, 92)
( 68,104)( 69,103)( 70,102)( 71,101)( 72,100)( 73, 99)( 74, 98)( 75, 97)
( 76, 96)( 77, 95)( 78, 94)(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)(157,197)(158,196)
(159,208)(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)
(167,200)(168,199)(169,198)(170,223)(171,222)(172,234)(173,233)(174,232)
(175,231)(176,230)(177,229)(178,228)(179,227)(180,226)(181,225)(182,224)
(183,210)(184,209)(185,221)(186,220)(187,219)(188,218)(189,217)(190,216)
(191,215)(192,214)(193,213)(194,212)(195,211);
s2 := 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);
s3 := 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);
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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
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