Polytope of Type {6,6,22}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,22}*1584a
Also Known As : {{6,6|2},{6,22|2}}. if this polytope has another 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, {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}*144a
12-fold quotients : {3,2,11}*132
18-fold quotients : {2,2,11}*88
33-fold quotients : {2,6,2}*48, {6,2,2}*48
66-fold quotients : {2,3,2}*24, {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)( 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, 45)( 13, 55)( 14, 54)( 15, 53)( 16, 52)
( 17, 51)( 18, 50)( 19, 49)( 20, 48)( 21, 47)( 22, 46)( 23, 56)( 24, 66)
( 25, 65)( 26, 64)( 27, 63)( 28, 62)( 29, 61)( 30, 60)( 31, 59)( 32, 58)
( 33, 57)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 79, 88)( 80, 87)
( 81, 86)( 82, 85)( 83, 84)( 90, 99)( 91, 98)( 92, 97)( 93, 96)( 94, 95)
(100,133)(101,143)(102,142)(103,141)(104,140)(105,139)(106,138)(107,137)
(108,136)(109,135)(110,134)(111,144)(112,154)(113,153)(114,152)(115,151)
(116,150)(117,149)(118,148)(119,147)(120,146)(121,145)(122,155)(123,165)
(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,157)
(132,156)(167,176)(168,175)(169,174)(170,173)(171,172)(178,187)(179,186)
(180,185)(181,184)(182,183)(189,198)(190,197)(191,196)(192,195)(193,194);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, 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, 45)( 13, 55)( 14, 54)( 15, 53)
( 16, 52)( 17, 51)( 18, 50)( 19, 49)( 20, 48)( 21, 47)( 22, 46)( 23, 56)
( 24, 66)( 25, 65)( 26, 64)( 27, 63)( 28, 62)( 29, 61)( 30, 60)( 31, 59)
( 32, 58)( 33, 57)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 79, 88)
( 80, 87)( 81, 86)( 82, 85)( 83, 84)( 90, 99)( 91, 98)( 92, 97)( 93, 96)
( 94, 95)(100,133)(101,143)(102,142)(103,141)(104,140)(105,139)(106,138)
(107,137)(108,136)(109,135)(110,134)(111,144)(112,154)(113,153)(114,152)
(115,151)(116,150)(117,149)(118,148)(119,147)(120,146)(121,145)(122,155)
(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)
(131,157)(132,156)(167,176)(168,175)(169,174)(170,173)(171,172)(178,187)
(179,186)(180,185)(181,184)(182,183)(189,198)(190,197)(191,196)(192,195)
(193,194);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
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