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Polytope of Type {22,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {22,18}*792
Also Known As : {22,18|2}. if this polytope has another name.
Group : SmallGroup(792,40)
Rank : 3
Schlafli Type : {22,18}
Number of vertices, edges, etc : 22, 198, 18
Order of s0s1s2 : 198
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {22,18,2} of size 1584
Vertex Figure Of :
   {2,22,18} of size 1584
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {22,6}*264
   9-fold quotients : {22,2}*88
   11-fold quotients : {2,18}*72
   18-fold quotients : {11,2}*44
   22-fold quotients : {2,9}*36
   33-fold quotients : {2,6}*24
   66-fold quotients : {2,3}*12
   99-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {22,36}*1584, {44,18}*1584a
Permutation Representation (GAP) :
s0 := (  4, 31)(  5, 32)(  6, 33)(  7, 28)(  8, 29)(  9, 30)( 10, 25)( 11, 26)
( 12, 27)( 13, 22)( 14, 23)( 15, 24)( 16, 19)( 17, 20)( 18, 21)( 37, 64)
( 38, 65)( 39, 66)( 40, 61)( 41, 62)( 42, 63)( 43, 58)( 44, 59)( 45, 60)
( 46, 55)( 47, 56)( 48, 57)( 49, 52)( 50, 53)( 51, 54)( 70, 97)( 71, 98)
( 72, 99)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)( 78, 93)( 79, 88)
( 80, 89)( 81, 90)( 82, 85)( 83, 86)( 84, 87)(103,130)(104,131)(105,132)
(106,127)(107,128)(108,129)(109,124)(110,125)(111,126)(112,121)(113,122)
(114,123)(115,118)(116,119)(117,120)(136,163)(137,164)(138,165)(139,160)
(140,161)(141,162)(142,157)(143,158)(144,159)(145,154)(146,155)(147,156)
(148,151)(149,152)(150,153)(169,196)(170,197)(171,198)(172,193)(173,194)
(174,195)(175,190)(176,191)(177,192)(178,187)(179,188)(180,189)(181,184)
(182,185)(183,186);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  7, 31)(  8, 33)(  9, 32)( 10, 28)( 11, 30)
( 12, 29)( 13, 25)( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 20, 21)
( 34, 72)( 35, 71)( 36, 70)( 37, 69)( 38, 68)( 39, 67)( 40, 99)( 41, 98)
( 42, 97)( 43, 96)( 44, 95)( 45, 94)( 46, 93)( 47, 92)( 48, 91)( 49, 90)
( 50, 89)( 51, 88)( 52, 87)( 53, 86)( 54, 85)( 55, 84)( 56, 83)( 57, 82)
( 58, 81)( 59, 80)( 60, 79)( 61, 78)( 62, 77)( 63, 76)( 64, 75)( 65, 74)
( 66, 73)(100,103)(101,105)(102,104)(106,130)(107,132)(108,131)(109,127)
(110,129)(111,128)(112,124)(113,126)(114,125)(115,121)(116,123)(117,122)
(119,120)(133,171)(134,170)(135,169)(136,168)(137,167)(138,166)(139,198)
(140,197)(141,196)(142,195)(143,194)(144,193)(145,192)(146,191)(147,190)
(148,189)(149,188)(150,187)(151,186)(152,185)(153,184)(154,183)(155,182)
(156,181)(157,180)(158,179)(159,178)(160,177)(161,176)(162,175)(163,174)
(164,173)(165,172);;
s2 := (  1,133)(  2,135)(  3,134)(  4,136)(  5,138)(  6,137)(  7,139)(  8,141)
(  9,140)( 10,142)( 11,144)( 12,143)( 13,145)( 14,147)( 15,146)( 16,148)
( 17,150)( 18,149)( 19,151)( 20,153)( 21,152)( 22,154)( 23,156)( 24,155)
( 25,157)( 26,159)( 27,158)( 28,160)( 29,162)( 30,161)( 31,163)( 32,165)
( 33,164)( 34,100)( 35,102)( 36,101)( 37,103)( 38,105)( 39,104)( 40,106)
( 41,108)( 42,107)( 43,109)( 44,111)( 45,110)( 46,112)( 47,114)( 48,113)
( 49,115)( 50,117)( 51,116)( 52,118)( 53,120)( 54,119)( 55,121)( 56,123)
( 57,122)( 58,124)( 59,126)( 60,125)( 61,127)( 62,129)( 63,128)( 64,130)
( 65,132)( 66,131)( 67,168)( 68,167)( 69,166)( 70,171)( 71,170)( 72,169)
( 73,174)( 74,173)( 75,172)( 76,177)( 77,176)( 78,175)( 79,180)( 80,179)
( 81,178)( 82,183)( 83,182)( 84,181)( 85,186)( 86,185)( 87,184)( 88,189)
( 89,188)( 90,187)( 91,192)( 92,191)( 93,190)( 94,195)( 95,194)( 96,193)
( 97,198)( 98,197)( 99,196);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(198)!(  4, 31)(  5, 32)(  6, 33)(  7, 28)(  8, 29)(  9, 30)( 10, 25)
( 11, 26)( 12, 27)( 13, 22)( 14, 23)( 15, 24)( 16, 19)( 17, 20)( 18, 21)
( 37, 64)( 38, 65)( 39, 66)( 40, 61)( 41, 62)( 42, 63)( 43, 58)( 44, 59)
( 45, 60)( 46, 55)( 47, 56)( 48, 57)( 49, 52)( 50, 53)( 51, 54)( 70, 97)
( 71, 98)( 72, 99)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)( 78, 93)
( 79, 88)( 80, 89)( 81, 90)( 82, 85)( 83, 86)( 84, 87)(103,130)(104,131)
(105,132)(106,127)(107,128)(108,129)(109,124)(110,125)(111,126)(112,121)
(113,122)(114,123)(115,118)(116,119)(117,120)(136,163)(137,164)(138,165)
(139,160)(140,161)(141,162)(142,157)(143,158)(144,159)(145,154)(146,155)
(147,156)(148,151)(149,152)(150,153)(169,196)(170,197)(171,198)(172,193)
(173,194)(174,195)(175,190)(176,191)(177,192)(178,187)(179,188)(180,189)
(181,184)(182,185)(183,186);
s1 := Sym(198)!(  1,  4)(  2,  6)(  3,  5)(  7, 31)(  8, 33)(  9, 32)( 10, 28)
( 11, 30)( 12, 29)( 13, 25)( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)
( 20, 21)( 34, 72)( 35, 71)( 36, 70)( 37, 69)( 38, 68)( 39, 67)( 40, 99)
( 41, 98)( 42, 97)( 43, 96)( 44, 95)( 45, 94)( 46, 93)( 47, 92)( 48, 91)
( 49, 90)( 50, 89)( 51, 88)( 52, 87)( 53, 86)( 54, 85)( 55, 84)( 56, 83)
( 57, 82)( 58, 81)( 59, 80)( 60, 79)( 61, 78)( 62, 77)( 63, 76)( 64, 75)
( 65, 74)( 66, 73)(100,103)(101,105)(102,104)(106,130)(107,132)(108,131)
(109,127)(110,129)(111,128)(112,124)(113,126)(114,125)(115,121)(116,123)
(117,122)(119,120)(133,171)(134,170)(135,169)(136,168)(137,167)(138,166)
(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)(145,192)(146,191)
(147,190)(148,189)(149,188)(150,187)(151,186)(152,185)(153,184)(154,183)
(155,182)(156,181)(157,180)(158,179)(159,178)(160,177)(161,176)(162,175)
(163,174)(164,173)(165,172);
s2 := Sym(198)!(  1,133)(  2,135)(  3,134)(  4,136)(  5,138)(  6,137)(  7,139)
(  8,141)(  9,140)( 10,142)( 11,144)( 12,143)( 13,145)( 14,147)( 15,146)
( 16,148)( 17,150)( 18,149)( 19,151)( 20,153)( 21,152)( 22,154)( 23,156)
( 24,155)( 25,157)( 26,159)( 27,158)( 28,160)( 29,162)( 30,161)( 31,163)
( 32,165)( 33,164)( 34,100)( 35,102)( 36,101)( 37,103)( 38,105)( 39,104)
( 40,106)( 41,108)( 42,107)( 43,109)( 44,111)( 45,110)( 46,112)( 47,114)
( 48,113)( 49,115)( 50,117)( 51,116)( 52,118)( 53,120)( 54,119)( 55,121)
( 56,123)( 57,122)( 58,124)( 59,126)( 60,125)( 61,127)( 62,129)( 63,128)
( 64,130)( 65,132)( 66,131)( 67,168)( 68,167)( 69,166)( 70,171)( 71,170)
( 72,169)( 73,174)( 74,173)( 75,172)( 76,177)( 77,176)( 78,175)( 79,180)
( 80,179)( 81,178)( 82,183)( 83,182)( 84,181)( 85,186)( 86,185)( 87,184)
( 88,189)( 89,188)( 90,187)( 91,192)( 92,191)( 93,190)( 94,195)( 95,194)
( 96,193)( 97,198)( 98,197)( 99,196);
poly := sub<Sym(198)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 >; 
 
References : None.
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