Polytope of Type {74,6,2}

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

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

to this polytope