Polytope of Type {2,34,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,34,14}*1904
if this polytope has a name.
Group : SmallGroup(1904,182)
Rank : 4
Schlafli Type : {2,34,14}
Number of vertices, edges, etc : 2, 34, 238, 14
Order of s₀s₁s₂s₃ : 238
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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) :
   7-fold quotients : {2,34,2}*272
   14-fold quotients : {2,17,2}*136
   17-fold quotients : {2,2,14}*112
   34-fold quotients : {2,2,7}*56
   119-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4, 19)(  5, 18)(  6, 17)(  7, 16)(  8, 15)(  9, 14)( 10, 13)( 11, 12)
( 21, 36)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 31)( 27, 30)( 28, 29)
( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)( 44, 47)( 45, 46)
( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)( 62, 63)
( 72, 87)( 73, 86)( 74, 85)( 75, 84)( 76, 83)( 77, 82)( 78, 81)( 79, 80)
( 89,104)( 90,103)( 91,102)( 92,101)( 93,100)( 94, 99)( 95, 98)( 96, 97)
(106,121)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)
(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)
(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)
(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)(164,165)
(174,189)(175,188)(176,187)(177,186)(178,185)(179,184)(180,183)(181,182)
(191,206)(192,205)(193,204)(194,203)(195,202)(196,201)(197,200)(198,199)
(208,223)(209,222)(210,221)(211,220)(212,219)(213,218)(214,217)(215,216)
(225,240)(226,239)(227,238)(228,237)(229,236)(230,235)(231,234)(232,233);;
s2 := (  3,  4)(  5, 19)(  6, 18)(  7, 17)(  8, 16)(  9, 15)( 10, 14)( 11, 13)
( 20,106)( 21,105)( 22,121)( 23,120)( 24,119)( 25,118)( 26,117)( 27,116)
( 28,115)( 29,114)( 30,113)( 31,112)( 32,111)( 33,110)( 34,109)( 35,108)
( 36,107)( 37, 89)( 38, 88)( 39,104)( 40,103)( 41,102)( 42,101)( 43,100)
( 44, 99)( 45, 98)( 46, 97)( 47, 96)( 48, 95)( 49, 94)( 50, 93)( 51, 92)
( 52, 91)( 53, 90)( 54, 72)( 55, 71)( 56, 87)( 57, 86)( 58, 85)( 59, 84)
( 60, 83)( 61, 82)( 62, 81)( 63, 80)( 64, 79)( 65, 78)( 66, 77)( 67, 76)
( 68, 75)( 69, 74)( 70, 73)(122,123)(124,138)(125,137)(126,136)(127,135)
(128,134)(129,133)(130,132)(139,225)(140,224)(141,240)(142,239)(143,238)
(144,237)(145,236)(146,235)(147,234)(148,233)(149,232)(150,231)(151,230)
(152,229)(153,228)(154,227)(155,226)(156,208)(157,207)(158,223)(159,222)
(160,221)(161,220)(162,219)(163,218)(164,217)(165,216)(166,215)(167,214)
(168,213)(169,212)(170,211)(171,210)(172,209)(173,191)(174,190)(175,206)
(176,205)(177,204)(178,203)(179,202)(180,201)(181,200)(182,199)(183,198)
(184,197)(185,196)(186,195)(187,194)(188,193)(189,192);;
s3 := (  3,139)(  4,140)(  5,141)(  6,142)(  7,143)(  8,144)(  9,145)( 10,146)
( 11,147)( 12,148)( 13,149)( 14,150)( 15,151)( 16,152)( 17,153)( 18,154)
( 19,155)( 20,122)( 21,123)( 22,124)( 23,125)( 24,126)( 25,127)( 26,128)
( 27,129)( 28,130)( 29,131)( 30,132)( 31,133)( 32,134)( 33,135)( 34,136)
( 35,137)( 36,138)( 37,224)( 38,225)( 39,226)( 40,227)( 41,228)( 42,229)
( 43,230)( 44,231)( 45,232)( 46,233)( 47,234)( 48,235)( 49,236)( 50,237)
( 51,238)( 52,239)( 53,240)( 54,207)( 55,208)( 56,209)( 57,210)( 58,211)
( 59,212)( 60,213)( 61,214)( 62,215)( 63,216)( 64,217)( 65,218)( 66,219)
( 67,220)( 68,221)( 69,222)( 70,223)( 71,190)( 72,191)( 73,192)( 74,193)
( 75,194)( 76,195)( 77,196)( 78,197)( 79,198)( 80,199)( 81,200)( 82,201)
( 83,202)( 84,203)( 85,204)( 86,205)( 87,206)( 88,173)( 89,174)( 90,175)
( 91,176)( 92,177)( 93,178)( 94,179)( 95,180)( 96,181)( 97,182)( 98,183)
( 99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,156)(106,157)
(107,158)(108,159)(109,160)(110,161)(111,162)(112,163)(113,164)(114,165)
(115,166)(116,167)(117,168)(118,169)(119,170)(120,171)(121,172);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*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, 
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(240)!(1,2);
s1 := Sym(240)!(  4, 19)(  5, 18)(  6, 17)(  7, 16)(  8, 15)(  9, 14)( 10, 13)
( 11, 12)( 21, 36)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 31)( 27, 30)
( 28, 29)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)( 44, 47)
( 45, 46)( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)
( 62, 63)( 72, 87)( 73, 86)( 74, 85)( 75, 84)( 76, 83)( 77, 82)( 78, 81)
( 79, 80)( 89,104)( 90,103)( 91,102)( 92,101)( 93,100)( 94, 99)( 95, 98)
( 96, 97)(106,121)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)
(113,114)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)
(130,131)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)
(147,148)(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)
(164,165)(174,189)(175,188)(176,187)(177,186)(178,185)(179,184)(180,183)
(181,182)(191,206)(192,205)(193,204)(194,203)(195,202)(196,201)(197,200)
(198,199)(208,223)(209,222)(210,221)(211,220)(212,219)(213,218)(214,217)
(215,216)(225,240)(226,239)(227,238)(228,237)(229,236)(230,235)(231,234)
(232,233);
s2 := Sym(240)!(  3,  4)(  5, 19)(  6, 18)(  7, 17)(  8, 16)(  9, 15)( 10, 14)
( 11, 13)( 20,106)( 21,105)( 22,121)( 23,120)( 24,119)( 25,118)( 26,117)
( 27,116)( 28,115)( 29,114)( 30,113)( 31,112)( 32,111)( 33,110)( 34,109)
( 35,108)( 36,107)( 37, 89)( 38, 88)( 39,104)( 40,103)( 41,102)( 42,101)
( 43,100)( 44, 99)( 45, 98)( 46, 97)( 47, 96)( 48, 95)( 49, 94)( 50, 93)
( 51, 92)( 52, 91)( 53, 90)( 54, 72)( 55, 71)( 56, 87)( 57, 86)( 58, 85)
( 59, 84)( 60, 83)( 61, 82)( 62, 81)( 63, 80)( 64, 79)( 65, 78)( 66, 77)
( 67, 76)( 68, 75)( 69, 74)( 70, 73)(122,123)(124,138)(125,137)(126,136)
(127,135)(128,134)(129,133)(130,132)(139,225)(140,224)(141,240)(142,239)
(143,238)(144,237)(145,236)(146,235)(147,234)(148,233)(149,232)(150,231)
(151,230)(152,229)(153,228)(154,227)(155,226)(156,208)(157,207)(158,223)
(159,222)(160,221)(161,220)(162,219)(163,218)(164,217)(165,216)(166,215)
(167,214)(168,213)(169,212)(170,211)(171,210)(172,209)(173,191)(174,190)
(175,206)(176,205)(177,204)(178,203)(179,202)(180,201)(181,200)(182,199)
(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192);
s3 := Sym(240)!(  3,139)(  4,140)(  5,141)(  6,142)(  7,143)(  8,144)(  9,145)
( 10,146)( 11,147)( 12,148)( 13,149)( 14,150)( 15,151)( 16,152)( 17,153)
( 18,154)( 19,155)( 20,122)( 21,123)( 22,124)( 23,125)( 24,126)( 25,127)
( 26,128)( 27,129)( 28,130)( 29,131)( 30,132)( 31,133)( 32,134)( 33,135)
( 34,136)( 35,137)( 36,138)( 37,224)( 38,225)( 39,226)( 40,227)( 41,228)
( 42,229)( 43,230)( 44,231)( 45,232)( 46,233)( 47,234)( 48,235)( 49,236)
( 50,237)( 51,238)( 52,239)( 53,240)( 54,207)( 55,208)( 56,209)( 57,210)
( 58,211)( 59,212)( 60,213)( 61,214)( 62,215)( 63,216)( 64,217)( 65,218)
( 66,219)( 67,220)( 68,221)( 69,222)( 70,223)( 71,190)( 72,191)( 73,192)
( 74,193)( 75,194)( 76,195)( 77,196)( 78,197)( 79,198)( 80,199)( 81,200)
( 82,201)( 83,202)( 84,203)( 85,204)( 86,205)( 87,206)( 88,173)( 89,174)
( 90,175)( 91,176)( 92,177)( 93,178)( 94,179)( 95,180)( 96,181)( 97,182)
( 98,183)( 99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,156)
(106,157)(107,158)(108,159)(109,160)(110,161)(111,162)(112,163)(113,164)
(114,165)(115,166)(116,167)(117,168)(118,169)(119,170)(120,171)(121,172);
poly := sub<Sym(240)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*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, 
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope