Polytope of Type {6,22,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,22,6}*1584
Also Known As : {{6,22|2},{22,6|2}}. if this polytope has another name.
Group : SmallGroup(1584,675)
Rank : 4
Schlafli Type : {6,22,6}
Number of vertices, edges, etc : 6, 66, 66, 6
Order of s0s1s2s3 : 66
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
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 : {2,22,6}*528, {6,22,2}*528
9-fold quotients : {2,22,2}*176
11-fold quotients : {6,2,6}*144
18-fold quotients : {2,11,2}*88
22-fold quotients : {3,2,6}*72, {6,2,3}*72
33-fold quotients : {2,2,6}*48, {6,2,2}*48
44-fold quotients : {3,2,3}*36
66-fold quotients : {2,2,3}*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, 22)( 3, 21)( 4, 20)( 5, 19)( 6, 18)( 7, 17)( 8, 16)
( 9, 15)( 10, 14)( 11, 13)( 24, 33)( 25, 32)( 26, 31)( 27, 30)( 28, 29)
( 34, 45)( 35, 55)( 36, 54)( 37, 53)( 38, 52)( 39, 51)( 40, 50)( 41, 49)
( 42, 48)( 43, 47)( 44, 46)( 57, 66)( 58, 65)( 59, 64)( 60, 63)( 61, 62)
( 67, 78)( 68, 88)( 69, 87)( 70, 86)( 71, 85)( 72, 84)( 73, 83)( 74, 82)
( 75, 81)( 76, 80)( 77, 79)( 90, 99)( 91, 98)( 92, 97)( 93, 96)( 94, 95)
(100,111)(101,121)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)
(108,114)(109,113)(110,112)(123,132)(124,131)(125,130)(126,129)(127,128)
(133,144)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)
(141,147)(142,146)(143,145)(156,165)(157,164)(158,163)(159,162)(160,161)
(166,177)(167,187)(168,186)(169,185)(170,184)(171,183)(172,182)(173,181)
(174,180)(175,179)(176,178)(189,198)(190,197)(191,196)(192,195)(193,194);;
s2 := ( 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, 68)
( 35, 67)( 36, 77)( 37, 76)( 38, 75)( 39, 74)( 40, 73)( 41, 72)( 42, 71)
( 43, 70)( 44, 69)( 45, 79)( 46, 78)( 47, 88)( 48, 87)( 49, 86)( 50, 85)
( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)( 56, 90)( 57, 89)( 58, 99)
( 59, 98)( 60, 97)( 61, 96)( 62, 95)( 63, 94)( 64, 93)( 65, 92)( 66, 91)
(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)(133,167)
(134,166)(135,176)(136,175)(137,174)(138,173)(139,172)(140,171)(141,170)
(142,169)(143,168)(144,178)(145,177)(146,187)(147,186)(148,185)(149,184)
(150,183)(151,182)(152,181)(153,180)(154,179)(155,189)(156,188)(157,198)
(158,197)(159,196)(160,195)(161,194)(162,193)(163,192)(164,191)(165,190);;
s3 := ( 1,133)( 2,134)( 3,135)( 4,136)( 5,137)( 6,138)( 7,139)( 8,140)
( 9,141)( 10,142)( 11,143)( 12,144)( 13,145)( 14,146)( 15,147)( 16,148)
( 17,149)( 18,150)( 19,151)( 20,152)( 21,153)( 22,154)( 23,155)( 24,156)
( 25,157)( 26,158)( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)( 32,164)
( 33,165)( 34,100)( 35,101)( 36,102)( 37,103)( 38,104)( 39,105)( 40,106)
( 41,107)( 42,108)( 43,109)( 44,110)( 45,111)( 46,112)( 47,113)( 48,114)
( 49,115)( 50,116)( 51,117)( 52,118)( 53,119)( 54,120)( 55,121)( 56,122)
( 57,123)( 58,124)( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)( 64,130)
( 65,131)( 66,132)( 67,166)( 68,167)( 69,168)( 70,169)( 71,170)( 72,171)
( 73,172)( 74,173)( 75,174)( 76,175)( 77,176)( 78,177)( 79,178)( 80,179)
( 81,180)( 82,181)( 83,182)( 84,183)( 85,184)( 86,185)( 87,186)( 88,187)
( 89,188)( 90,189)( 91,190)( 92,191)( 93,192)( 94,193)( 95,194)( 96,195)
( 97,196)( 98,197)( 99,198);;
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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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(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, 22)( 3, 21)( 4, 20)( 5, 19)( 6, 18)( 7, 17)
( 8, 16)( 9, 15)( 10, 14)( 11, 13)( 24, 33)( 25, 32)( 26, 31)( 27, 30)
( 28, 29)( 34, 45)( 35, 55)( 36, 54)( 37, 53)( 38, 52)( 39, 51)( 40, 50)
( 41, 49)( 42, 48)( 43, 47)( 44, 46)( 57, 66)( 58, 65)( 59, 64)( 60, 63)
( 61, 62)( 67, 78)( 68, 88)( 69, 87)( 70, 86)( 71, 85)( 72, 84)( 73, 83)
( 74, 82)( 75, 81)( 76, 80)( 77, 79)( 90, 99)( 91, 98)( 92, 97)( 93, 96)
( 94, 95)(100,111)(101,121)(102,120)(103,119)(104,118)(105,117)(106,116)
(107,115)(108,114)(109,113)(110,112)(123,132)(124,131)(125,130)(126,129)
(127,128)(133,144)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)
(140,148)(141,147)(142,146)(143,145)(156,165)(157,164)(158,163)(159,162)
(160,161)(166,177)(167,187)(168,186)(169,185)(170,184)(171,183)(172,182)
(173,181)(174,180)(175,179)(176,178)(189,198)(190,197)(191,196)(192,195)
(193,194);
s2 := 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, 68)( 35, 67)( 36, 77)( 37, 76)( 38, 75)( 39, 74)( 40, 73)( 41, 72)
( 42, 71)( 43, 70)( 44, 69)( 45, 79)( 46, 78)( 47, 88)( 48, 87)( 49, 86)
( 50, 85)( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)( 56, 90)( 57, 89)
( 58, 99)( 59, 98)( 60, 97)( 61, 96)( 62, 95)( 63, 94)( 64, 93)( 65, 92)
( 66, 91)(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)
(133,167)(134,166)(135,176)(136,175)(137,174)(138,173)(139,172)(140,171)
(141,170)(142,169)(143,168)(144,178)(145,177)(146,187)(147,186)(148,185)
(149,184)(150,183)(151,182)(152,181)(153,180)(154,179)(155,189)(156,188)
(157,198)(158,197)(159,196)(160,195)(161,194)(162,193)(163,192)(164,191)
(165,190);
s3 := Sym(198)!( 1,133)( 2,134)( 3,135)( 4,136)( 5,137)( 6,138)( 7,139)
( 8,140)( 9,141)( 10,142)( 11,143)( 12,144)( 13,145)( 14,146)( 15,147)
( 16,148)( 17,149)( 18,150)( 19,151)( 20,152)( 21,153)( 22,154)( 23,155)
( 24,156)( 25,157)( 26,158)( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)
( 32,164)( 33,165)( 34,100)( 35,101)( 36,102)( 37,103)( 38,104)( 39,105)
( 40,106)( 41,107)( 42,108)( 43,109)( 44,110)( 45,111)( 46,112)( 47,113)
( 48,114)( 49,115)( 50,116)( 51,117)( 52,118)( 53,119)( 54,120)( 55,121)
( 56,122)( 57,123)( 58,124)( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)
( 64,130)( 65,131)( 66,132)( 67,166)( 68,167)( 69,168)( 70,169)( 71,170)
( 72,171)( 73,172)( 74,173)( 75,174)( 76,175)( 77,176)( 78,177)( 79,178)
( 80,179)( 81,180)( 82,181)( 83,182)( 84,183)( 85,184)( 86,185)( 87,186)
( 88,187)( 89,188)( 90,189)( 91,190)( 92,191)( 93,192)( 94,193)( 95,194)
( 96,195)( 97,196)( 98,197)( 99,198);
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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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.
to this polytope