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Polytope of Type {6,74}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,74}*888
Also Known As : {6,74|2}. if this polytope has another name.
Group : SmallGroup(888,54)
Rank : 3
Schlafli Type : {6,74}
Number of vertices, edges, etc : 6, 222, 74
Order of s0s1s2 : 222
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,74,2} of size 1776
Vertex Figure Of :
   {2,6,74} of size 1776
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,74}*296
   6-fold quotients : {2,37}*148
   37-fold quotients : {6,2}*24
   74-fold quotients : {3,2}*12
   111-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,74}*1776, {6,148}*1776a
Permutation Representation (GAP) :
s0 := ( 38, 75)( 39, 76)( 40, 77)( 41, 78)( 42, 79)( 43, 80)( 44, 81)( 45, 82)
( 46, 83)( 47, 84)( 48, 85)( 49, 86)( 50, 87)( 51, 88)( 52, 89)( 53, 90)
( 54, 91)( 55, 92)( 56, 93)( 57, 94)( 58, 95)( 59, 96)( 60, 97)( 61, 98)
( 62, 99)( 63,100)( 64,101)( 65,102)( 66,103)( 67,104)( 68,105)( 69,106)
( 70,107)( 71,108)( 72,109)( 73,110)( 74,111)(149,186)(150,187)(151,188)
(152,189)(153,190)(154,191)(155,192)(156,193)(157,194)(158,195)(159,196)
(160,197)(161,198)(162,199)(163,200)(164,201)(165,202)(166,203)(167,204)
(168,205)(169,206)(170,207)(171,208)(172,209)(173,210)(174,211)(175,212)
(176,213)(177,214)(178,215)(179,216)(180,217)(181,218)(182,219)(183,220)
(184,221)(185,222);;
s1 := (  1, 38)(  2, 74)(  3, 73)(  4, 72)(  5, 71)(  6, 70)(  7, 69)(  8, 68)
(  9, 67)( 10, 66)( 11, 65)( 12, 64)( 13, 63)( 14, 62)( 15, 61)( 16, 60)
( 17, 59)( 18, 58)( 19, 57)( 20, 56)( 21, 55)( 22, 54)( 23, 53)( 24, 52)
( 25, 51)( 26, 50)( 27, 49)( 28, 48)( 29, 47)( 30, 46)( 31, 45)( 32, 44)
( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 37, 39)( 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)(112,149)
(113,185)(114,184)(115,183)(116,182)(117,181)(118,180)(119,179)(120,178)
(121,177)(122,176)(123,175)(124,174)(125,173)(126,172)(127,171)(128,170)
(129,169)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)(136,162)
(137,161)(138,160)(139,159)(140,158)(141,157)(142,156)(143,155)(144,154)
(145,153)(146,152)(147,151)(148,150)(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);;
s2 := (  1,113)(  2,112)(  3,148)(  4,147)(  5,146)(  6,145)(  7,144)(  8,143)
(  9,142)( 10,141)( 11,140)( 12,139)( 13,138)( 14,137)( 15,136)( 16,135)
( 17,134)( 18,133)( 19,132)( 20,131)( 21,130)( 22,129)( 23,128)( 24,127)
( 25,126)( 26,125)( 27,124)( 28,123)( 29,122)( 30,121)( 31,120)( 32,119)
( 33,118)( 34,117)( 35,116)( 36,115)( 37,114)( 38,150)( 39,149)( 40,185)
( 41,184)( 42,183)( 43,182)( 44,181)( 45,180)( 46,179)( 47,178)( 48,177)
( 49,176)( 50,175)( 51,174)( 52,173)( 53,172)( 54,171)( 55,170)( 56,169)
( 57,168)( 58,167)( 59,166)( 60,165)( 61,164)( 62,163)( 63,162)( 64,161)
( 65,160)( 66,159)( 67,158)( 68,157)( 69,156)( 70,155)( 71,154)( 72,153)
( 73,152)( 74,151)( 75,187)( 76,186)( 77,222)( 78,221)( 79,220)( 80,219)
( 81,218)( 82,217)( 83,216)( 84,215)( 85,214)( 86,213)( 87,212)( 88,211)
( 89,210)( 90,209)( 91,208)( 92,207)( 93,206)( 94,205)( 95,204)( 96,203)
( 97,202)( 98,201)( 99,200)(100,199)(101,198)(102,197)(103,196)(104,195)
(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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(222)!( 38, 75)( 39, 76)( 40, 77)( 41, 78)( 42, 79)( 43, 80)( 44, 81)
( 45, 82)( 46, 83)( 47, 84)( 48, 85)( 49, 86)( 50, 87)( 51, 88)( 52, 89)
( 53, 90)( 54, 91)( 55, 92)( 56, 93)( 57, 94)( 58, 95)( 59, 96)( 60, 97)
( 61, 98)( 62, 99)( 63,100)( 64,101)( 65,102)( 66,103)( 67,104)( 68,105)
( 69,106)( 70,107)( 71,108)( 72,109)( 73,110)( 74,111)(149,186)(150,187)
(151,188)(152,189)(153,190)(154,191)(155,192)(156,193)(157,194)(158,195)
(159,196)(160,197)(161,198)(162,199)(163,200)(164,201)(165,202)(166,203)
(167,204)(168,205)(169,206)(170,207)(171,208)(172,209)(173,210)(174,211)
(175,212)(176,213)(177,214)(178,215)(179,216)(180,217)(181,218)(182,219)
(183,220)(184,221)(185,222);
s1 := Sym(222)!(  1, 38)(  2, 74)(  3, 73)(  4, 72)(  5, 71)(  6, 70)(  7, 69)
(  8, 68)(  9, 67)( 10, 66)( 11, 65)( 12, 64)( 13, 63)( 14, 62)( 15, 61)
( 16, 60)( 17, 59)( 18, 58)( 19, 57)( 20, 56)( 21, 55)( 22, 54)( 23, 53)
( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 28, 48)( 29, 47)( 30, 46)( 31, 45)
( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 37, 39)( 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)
(112,149)(113,185)(114,184)(115,183)(116,182)(117,181)(118,180)(119,179)
(120,178)(121,177)(122,176)(123,175)(124,174)(125,173)(126,172)(127,171)
(128,170)(129,169)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)
(136,162)(137,161)(138,160)(139,159)(140,158)(141,157)(142,156)(143,155)
(144,154)(145,153)(146,152)(147,151)(148,150)(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);
s2 := Sym(222)!(  1,113)(  2,112)(  3,148)(  4,147)(  5,146)(  6,145)(  7,144)
(  8,143)(  9,142)( 10,141)( 11,140)( 12,139)( 13,138)( 14,137)( 15,136)
( 16,135)( 17,134)( 18,133)( 19,132)( 20,131)( 21,130)( 22,129)( 23,128)
( 24,127)( 25,126)( 26,125)( 27,124)( 28,123)( 29,122)( 30,121)( 31,120)
( 32,119)( 33,118)( 34,117)( 35,116)( 36,115)( 37,114)( 38,150)( 39,149)
( 40,185)( 41,184)( 42,183)( 43,182)( 44,181)( 45,180)( 46,179)( 47,178)
( 48,177)( 49,176)( 50,175)( 51,174)( 52,173)( 53,172)( 54,171)( 55,170)
( 56,169)( 57,168)( 58,167)( 59,166)( 60,165)( 61,164)( 62,163)( 63,162)
( 64,161)( 65,160)( 66,159)( 67,158)( 68,157)( 69,156)( 70,155)( 71,154)
( 72,153)( 73,152)( 74,151)( 75,187)( 76,186)( 77,222)( 78,221)( 79,220)
( 80,219)( 81,218)( 82,217)( 83,216)( 84,215)( 85,214)( 86,213)( 87,212)
( 88,211)( 89,210)( 90,209)( 91,208)( 92,207)( 93,206)( 94,205)( 95,204)
( 96,203)( 97,202)( 98,201)( 99,200)(100,199)(101,198)(102,197)(103,196)
(104,195)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188);
poly := sub<Sym(222)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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 >; 
 
References : None.
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