Polytope of Type {6,6,22}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,22}*1584c
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) :
   2-fold quotients : {3,6,22}*792
   3-fold quotients : {6,2,22}*528
   6-fold quotients : {3,2,22}*264, {6,2,11}*264
   9-fold quotients : {2,2,22}*176
   11-fold quotients : {6,6,2}*144c
   12-fold quotients : {3,2,11}*132
   18-fold quotients : {2,2,11}*88
   22-fold quotients : {3,6,2}*72
   33-fold quotients : {6,2,2}*48
   66-fold quotients : {3,2,2}*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)( 34, 67)( 35, 68)( 36, 69)( 37, 70)( 38, 71)
( 39, 72)( 40, 73)( 41, 74)( 42, 75)( 43, 76)( 44, 77)( 45, 89)( 46, 90)
( 47, 91)( 48, 92)( 49, 93)( 50, 94)( 51, 95)( 52, 96)( 53, 97)( 54, 98)
( 55, 99)( 56, 78)( 57, 79)( 58, 80)( 59, 81)( 60, 82)( 61, 83)( 62, 84)
( 63, 85)( 64, 86)( 65, 87)( 66, 88)(111,122)(112,123)(113,124)(114,125)
(115,126)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132)(133,166)
(134,167)(135,168)(136,169)(137,170)(138,171)(139,172)(140,173)(141,174)
(142,175)(143,176)(144,188)(145,189)(146,190)(147,191)(148,192)(149,193)
(150,194)(151,195)(152,196)(153,197)(154,198)(155,177)(156,178)(157,179)
(158,180)(159,181)(160,182)(161,183)(162,184)(163,185)(164,186)(165,187);;
s1 := (  1,144)(  2,145)(  3,146)(  4,147)(  5,148)(  6,149)(  7,150)(  8,151)
(  9,152)( 10,153)( 11,154)( 12,133)( 13,134)( 14,135)( 15,136)( 16,137)
( 17,138)( 18,139)( 19,140)( 20,141)( 21,142)( 22,143)( 23,155)( 24,156)
( 25,157)( 26,158)( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)( 32,164)
( 33,165)( 34,111)( 35,112)( 36,113)( 37,114)( 38,115)( 39,116)( 40,117)
( 41,118)( 42,119)( 43,120)( 44,121)( 45,100)( 46,101)( 47,102)( 48,103)
( 49,104)( 50,105)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)( 56,122)
( 57,123)( 58,124)( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)( 64,130)
( 65,131)( 66,132)( 67,177)( 68,178)( 69,179)( 70,180)( 71,181)( 72,182)
( 73,183)( 74,184)( 75,185)( 76,186)( 77,187)( 78,166)( 79,167)( 80,168)
( 81,169)( 82,170)( 83,171)( 84,172)( 85,173)( 86,174)( 87,175)( 88,176)
( 89,188)( 90,189)( 91,190)( 92,191)( 93,192)( 94,193)( 95,194)( 96,195)
( 97,196)( 98,197)( 99,198);;
s2 := (  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 12, 23)( 13, 33)( 14, 32)
( 15, 31)( 16, 30)( 17, 29)( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)
( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 45, 56)( 46, 66)( 47, 65)
( 48, 64)( 49, 63)( 50, 62)( 51, 61)( 52, 60)( 53, 59)( 54, 58)( 55, 57)
( 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)
(101,110)(102,109)(103,108)(104,107)(105,106)(111,122)(112,132)(113,131)
(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)
(134,143)(135,142)(136,141)(137,140)(138,139)(144,155)(145,165)(146,164)
(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(154,156)
(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,  2)(  3, 11)(  4, 10)(  5,  9)(  6,  8)( 12, 13)( 14, 22)( 15, 21)
( 16, 20)( 17, 19)( 23, 24)( 25, 33)( 26, 32)( 27, 31)( 28, 30)( 34, 35)
( 36, 44)( 37, 43)( 38, 42)( 39, 41)( 45, 46)( 47, 55)( 48, 54)( 49, 53)
( 50, 52)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)( 67, 68)( 69, 77)
( 70, 76)( 71, 75)( 72, 74)( 78, 79)( 80, 88)( 81, 87)( 82, 86)( 83, 85)
( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(100,101)(102,110)(103,109)
(104,108)(105,107)(111,112)(113,121)(114,120)(115,119)(116,118)(122,123)
(124,132)(125,131)(126,130)(127,129)(133,134)(135,143)(136,142)(137,141)
(138,140)(144,145)(146,154)(147,153)(148,152)(149,151)(155,156)(157,165)
(158,164)(159,163)(160,162)(166,167)(168,176)(169,175)(170,174)(171,173)
(177,178)(179,187)(180,186)(181,185)(182,184)(188,189)(190,198)(191,197)
(192,196)(193,195);;
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*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
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)( 34, 67)( 35, 68)( 36, 69)( 37, 70)
( 38, 71)( 39, 72)( 40, 73)( 41, 74)( 42, 75)( 43, 76)( 44, 77)( 45, 89)
( 46, 90)( 47, 91)( 48, 92)( 49, 93)( 50, 94)( 51, 95)( 52, 96)( 53, 97)
( 54, 98)( 55, 99)( 56, 78)( 57, 79)( 58, 80)( 59, 81)( 60, 82)( 61, 83)
( 62, 84)( 63, 85)( 64, 86)( 65, 87)( 66, 88)(111,122)(112,123)(113,124)
(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132)
(133,166)(134,167)(135,168)(136,169)(137,170)(138,171)(139,172)(140,173)
(141,174)(142,175)(143,176)(144,188)(145,189)(146,190)(147,191)(148,192)
(149,193)(150,194)(151,195)(152,196)(153,197)(154,198)(155,177)(156,178)
(157,179)(158,180)(159,181)(160,182)(161,183)(162,184)(163,185)(164,186)
(165,187);
s1 := Sym(198)!(  1,144)(  2,145)(  3,146)(  4,147)(  5,148)(  6,149)(  7,150)
(  8,151)(  9,152)( 10,153)( 11,154)( 12,133)( 13,134)( 14,135)( 15,136)
( 16,137)( 17,138)( 18,139)( 19,140)( 20,141)( 21,142)( 22,143)( 23,155)
( 24,156)( 25,157)( 26,158)( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)
( 32,164)( 33,165)( 34,111)( 35,112)( 36,113)( 37,114)( 38,115)( 39,116)
( 40,117)( 41,118)( 42,119)( 43,120)( 44,121)( 45,100)( 46,101)( 47,102)
( 48,103)( 49,104)( 50,105)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)
( 56,122)( 57,123)( 58,124)( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)
( 64,130)( 65,131)( 66,132)( 67,177)( 68,178)( 69,179)( 70,180)( 71,181)
( 72,182)( 73,183)( 74,184)( 75,185)( 76,186)( 77,187)( 78,166)( 79,167)
( 80,168)( 81,169)( 82,170)( 83,171)( 84,172)( 85,173)( 86,174)( 87,175)
( 88,176)( 89,188)( 90,189)( 91,190)( 92,191)( 93,192)( 94,193)( 95,194)
( 96,195)( 97,196)( 98,197)( 99,198);
s2 := Sym(198)!(  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 12, 23)( 13, 33)
( 14, 32)( 15, 31)( 16, 30)( 17, 29)( 18, 28)( 19, 27)( 20, 26)( 21, 25)
( 22, 24)( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 45, 56)( 46, 66)
( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 51, 61)( 52, 60)( 53, 59)( 54, 58)
( 55, 57)( 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)(101,110)(102,109)(103,108)(104,107)(105,106)(111,122)(112,132)
(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)
(121,123)(134,143)(135,142)(136,141)(137,140)(138,139)(144,155)(145,165)
(146,164)(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)
(154,156)(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,  2)(  3, 11)(  4, 10)(  5,  9)(  6,  8)( 12, 13)( 14, 22)
( 15, 21)( 16, 20)( 17, 19)( 23, 24)( 25, 33)( 26, 32)( 27, 31)( 28, 30)
( 34, 35)( 36, 44)( 37, 43)( 38, 42)( 39, 41)( 45, 46)( 47, 55)( 48, 54)
( 49, 53)( 50, 52)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)( 67, 68)
( 69, 77)( 70, 76)( 71, 75)( 72, 74)( 78, 79)( 80, 88)( 81, 87)( 82, 86)
( 83, 85)( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(100,101)(102,110)
(103,109)(104,108)(105,107)(111,112)(113,121)(114,120)(115,119)(116,118)
(122,123)(124,132)(125,131)(126,130)(127,129)(133,134)(135,143)(136,142)
(137,141)(138,140)(144,145)(146,154)(147,153)(148,152)(149,151)(155,156)
(157,165)(158,164)(159,163)(160,162)(166,167)(168,176)(169,175)(170,174)
(171,173)(177,178)(179,187)(180,186)(181,185)(182,184)(188,189)(190,198)
(191,197)(192,196)(193,195);
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
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