Polytope of Type {4,6,22}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,22}*1056b
if this polytope has a name.
Group : SmallGroup(1056,1015)
Rank : 4
Schlafli Type : {4,6,22}
Number of vertices, edges, etc : 4, 12, 66, 22
Order of s0s1s2s3 : 66
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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) :
11-fold quotients : {4,6,2}*96c
22-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264);;
s1 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 93)
( 50, 94)( 51, 96)( 52, 95)( 53, 97)( 54, 98)( 55,100)( 56, 99)( 57,101)
( 58,102)( 59,104)( 60,103)( 61,105)( 62,106)( 63,108)( 64,107)( 65,109)
( 66,110)( 67,112)( 68,111)( 69,113)( 70,114)( 71,116)( 72,115)( 73,117)
( 74,118)( 75,120)( 76,119)( 77,121)( 78,122)( 79,124)( 80,123)( 81,125)
( 82,126)( 83,128)( 84,127)( 85,129)( 86,130)( 87,132)( 88,131)(135,136)
(139,140)(143,144)(147,148)(151,152)(155,156)(159,160)(163,164)(167,168)
(171,172)(175,176)(177,221)(178,222)(179,224)(180,223)(181,225)(182,226)
(183,228)(184,227)(185,229)(186,230)(187,232)(188,231)(189,233)(190,234)
(191,236)(192,235)(193,237)(194,238)(195,240)(196,239)(197,241)(198,242)
(199,244)(200,243)(201,245)(202,246)(203,248)(204,247)(205,249)(206,250)
(207,252)(208,251)(209,253)(210,254)(211,256)(212,255)(213,257)(214,258)
(215,260)(216,259)(217,261)(218,262)(219,264)(220,263);;
s2 := ( 1, 45)( 2, 48)( 3, 47)( 4, 46)( 5, 85)( 6, 88)( 7, 87)( 8, 86)
( 9, 81)( 10, 84)( 11, 83)( 12, 82)( 13, 77)( 14, 80)( 15, 79)( 16, 78)
( 17, 73)( 18, 76)( 19, 75)( 20, 74)( 21, 69)( 22, 72)( 23, 71)( 24, 70)
( 25, 65)( 26, 68)( 27, 67)( 28, 66)( 29, 61)( 30, 64)( 31, 63)( 32, 62)
( 33, 57)( 34, 60)( 35, 59)( 36, 58)( 37, 53)( 38, 56)( 39, 55)( 40, 54)
( 41, 49)( 42, 52)( 43, 51)( 44, 50)( 90, 92)( 93,129)( 94,132)( 95,131)
( 96,130)( 97,125)( 98,128)( 99,127)(100,126)(101,121)(102,124)(103,123)
(104,122)(105,117)(106,120)(107,119)(108,118)(109,113)(110,116)(111,115)
(112,114)(133,177)(134,180)(135,179)(136,178)(137,217)(138,220)(139,219)
(140,218)(141,213)(142,216)(143,215)(144,214)(145,209)(146,212)(147,211)
(148,210)(149,205)(150,208)(151,207)(152,206)(153,201)(154,204)(155,203)
(156,202)(157,197)(158,200)(159,199)(160,198)(161,193)(162,196)(163,195)
(164,194)(165,189)(166,192)(167,191)(168,190)(169,185)(170,188)(171,187)
(172,186)(173,181)(174,184)(175,183)(176,182)(222,224)(225,261)(226,264)
(227,263)(228,262)(229,257)(230,260)(231,259)(232,258)(233,253)(234,256)
(235,255)(236,254)(237,249)(238,252)(239,251)(240,250)(241,245)(242,248)
(243,247)(244,246);;
s3 := ( 1,137)( 2,138)( 3,139)( 4,140)( 5,133)( 6,134)( 7,135)( 8,136)
( 9,173)( 10,174)( 11,175)( 12,176)( 13,169)( 14,170)( 15,171)( 16,172)
( 17,165)( 18,166)( 19,167)( 20,168)( 21,161)( 22,162)( 23,163)( 24,164)
( 25,157)( 26,158)( 27,159)( 28,160)( 29,153)( 30,154)( 31,155)( 32,156)
( 33,149)( 34,150)( 35,151)( 36,152)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,141)( 42,142)( 43,143)( 44,144)( 45,181)( 46,182)( 47,183)( 48,184)
( 49,177)( 50,178)( 51,179)( 52,180)( 53,217)( 54,218)( 55,219)( 56,220)
( 57,213)( 58,214)( 59,215)( 60,216)( 61,209)( 62,210)( 63,211)( 64,212)
( 65,205)( 66,206)( 67,207)( 68,208)( 69,201)( 70,202)( 71,203)( 72,204)
( 73,197)( 74,198)( 75,199)( 76,200)( 77,193)( 78,194)( 79,195)( 80,196)
( 81,189)( 82,190)( 83,191)( 84,192)( 85,185)( 86,186)( 87,187)( 88,188)
( 89,225)( 90,226)( 91,227)( 92,228)( 93,221)( 94,222)( 95,223)( 96,224)
( 97,261)( 98,262)( 99,263)(100,264)(101,257)(102,258)(103,259)(104,260)
(105,253)(106,254)(107,255)(108,256)(109,249)(110,250)(111,251)(112,252)
(113,245)(114,246)(115,247)(116,248)(117,241)(118,242)(119,243)(120,244)
(121,237)(122,238)(123,239)(124,240)(125,233)(126,234)(127,235)(128,236)
(129,229)(130,230)(131,231)(132,232);;
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*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*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(264)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264);
s1 := Sym(264)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 45, 89)( 46, 90)( 47, 92)( 48, 91)
( 49, 93)( 50, 94)( 51, 96)( 52, 95)( 53, 97)( 54, 98)( 55,100)( 56, 99)
( 57,101)( 58,102)( 59,104)( 60,103)( 61,105)( 62,106)( 63,108)( 64,107)
( 65,109)( 66,110)( 67,112)( 68,111)( 69,113)( 70,114)( 71,116)( 72,115)
( 73,117)( 74,118)( 75,120)( 76,119)( 77,121)( 78,122)( 79,124)( 80,123)
( 81,125)( 82,126)( 83,128)( 84,127)( 85,129)( 86,130)( 87,132)( 88,131)
(135,136)(139,140)(143,144)(147,148)(151,152)(155,156)(159,160)(163,164)
(167,168)(171,172)(175,176)(177,221)(178,222)(179,224)(180,223)(181,225)
(182,226)(183,228)(184,227)(185,229)(186,230)(187,232)(188,231)(189,233)
(190,234)(191,236)(192,235)(193,237)(194,238)(195,240)(196,239)(197,241)
(198,242)(199,244)(200,243)(201,245)(202,246)(203,248)(204,247)(205,249)
(206,250)(207,252)(208,251)(209,253)(210,254)(211,256)(212,255)(213,257)
(214,258)(215,260)(216,259)(217,261)(218,262)(219,264)(220,263);
s2 := Sym(264)!( 1, 45)( 2, 48)( 3, 47)( 4, 46)( 5, 85)( 6, 88)( 7, 87)
( 8, 86)( 9, 81)( 10, 84)( 11, 83)( 12, 82)( 13, 77)( 14, 80)( 15, 79)
( 16, 78)( 17, 73)( 18, 76)( 19, 75)( 20, 74)( 21, 69)( 22, 72)( 23, 71)
( 24, 70)( 25, 65)( 26, 68)( 27, 67)( 28, 66)( 29, 61)( 30, 64)( 31, 63)
( 32, 62)( 33, 57)( 34, 60)( 35, 59)( 36, 58)( 37, 53)( 38, 56)( 39, 55)
( 40, 54)( 41, 49)( 42, 52)( 43, 51)( 44, 50)( 90, 92)( 93,129)( 94,132)
( 95,131)( 96,130)( 97,125)( 98,128)( 99,127)(100,126)(101,121)(102,124)
(103,123)(104,122)(105,117)(106,120)(107,119)(108,118)(109,113)(110,116)
(111,115)(112,114)(133,177)(134,180)(135,179)(136,178)(137,217)(138,220)
(139,219)(140,218)(141,213)(142,216)(143,215)(144,214)(145,209)(146,212)
(147,211)(148,210)(149,205)(150,208)(151,207)(152,206)(153,201)(154,204)
(155,203)(156,202)(157,197)(158,200)(159,199)(160,198)(161,193)(162,196)
(163,195)(164,194)(165,189)(166,192)(167,191)(168,190)(169,185)(170,188)
(171,187)(172,186)(173,181)(174,184)(175,183)(176,182)(222,224)(225,261)
(226,264)(227,263)(228,262)(229,257)(230,260)(231,259)(232,258)(233,253)
(234,256)(235,255)(236,254)(237,249)(238,252)(239,251)(240,250)(241,245)
(242,248)(243,247)(244,246);
s3 := Sym(264)!( 1,137)( 2,138)( 3,139)( 4,140)( 5,133)( 6,134)( 7,135)
( 8,136)( 9,173)( 10,174)( 11,175)( 12,176)( 13,169)( 14,170)( 15,171)
( 16,172)( 17,165)( 18,166)( 19,167)( 20,168)( 21,161)( 22,162)( 23,163)
( 24,164)( 25,157)( 26,158)( 27,159)( 28,160)( 29,153)( 30,154)( 31,155)
( 32,156)( 33,149)( 34,150)( 35,151)( 36,152)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,141)( 42,142)( 43,143)( 44,144)( 45,181)( 46,182)( 47,183)
( 48,184)( 49,177)( 50,178)( 51,179)( 52,180)( 53,217)( 54,218)( 55,219)
( 56,220)( 57,213)( 58,214)( 59,215)( 60,216)( 61,209)( 62,210)( 63,211)
( 64,212)( 65,205)( 66,206)( 67,207)( 68,208)( 69,201)( 70,202)( 71,203)
( 72,204)( 73,197)( 74,198)( 75,199)( 76,200)( 77,193)( 78,194)( 79,195)
( 80,196)( 81,189)( 82,190)( 83,191)( 84,192)( 85,185)( 86,186)( 87,187)
( 88,188)( 89,225)( 90,226)( 91,227)( 92,228)( 93,221)( 94,222)( 95,223)
( 96,224)( 97,261)( 98,262)( 99,263)(100,264)(101,257)(102,258)(103,259)
(104,260)(105,253)(106,254)(107,255)(108,256)(109,249)(110,250)(111,251)
(112,252)(113,245)(114,246)(115,247)(116,248)(117,241)(118,242)(119,243)
(120,244)(121,237)(122,238)(123,239)(124,240)(125,233)(126,234)(127,235)
(128,236)(129,229)(130,230)(131,231)(132,232);
poly := sub<Sym(264)|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*s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s0*s1*s2*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