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