Polytope of Type {6,6,22}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,22}*1584b
if this polytope has a name.
Group : SmallGroup(1584,675)
Rank : 4
Schlafli Type : {6,6,22}
Number of vertices, edges, etc : 6, 18, 66, 22
Order of s₀s₁s₂s₃ : 66
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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 : {2,6,22}*528
   9-fold quotients : {2,2,22}*176
   11-fold quotients : {6,6,2}*144b
   18-fold quotients : {2,2,11}*88
   22-fold quotients : {6,3,2}*72
   33-fold quotients : {2,6,2}*48
   66-fold quotients : {2,3,2}*24
   99-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope