Polytope of Type {2,74,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,74,6}*1776
if this polytope has a name.
Group : SmallGroup(1776,251)
Rank : 4
Schlafli Type : {2,74,6}
Number of vertices, edges, etc : 2, 74, 222, 6
Order of s₀s₁s₂s₃ : 222
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) :
   3-fold quotients : {2,74,2}*592
   6-fold quotients : {2,37,2}*296
   37-fold quotients : {2,2,6}*48
   74-fold quotients : {2,2,3}*24
   111-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope