Polytope of Type {6,6,15}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,15}*1080a
if this polytope has a name.
Group : SmallGroup(1080,337)
Rank : 4
Schlafli Type : {6,6,15}
Number of vertices, edges, etc : 6, 18, 45, 15
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 6
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 : {3,6,15}*540
3-fold quotients : {6,2,15}*360
5-fold quotients : {6,6,3}*216a
6-fold quotients : {3,2,15}*180
9-fold quotients : {6,2,5}*120, {2,2,15}*120
10-fold quotients : {3,6,3}*108
15-fold quotients : {6,2,3}*72
18-fold quotients : {3,2,5}*60
27-fold quotients : {2,2,5}*40
30-fold quotients : {3,2,3}*36
45-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 46, 91)
( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98)
( 55,100)( 56,102)( 57,101)( 58,103)( 59,105)( 60,104)( 61,106)( 62,108)
( 63,107)( 64,109)( 65,111)( 66,110)( 67,112)( 68,114)( 69,113)( 70,115)
( 71,117)( 72,116)( 73,118)( 74,120)( 75,119)( 76,121)( 77,123)( 78,122)
( 79,124)( 80,126)( 81,125)( 82,127)( 83,129)( 84,128)( 85,130)( 86,132)
( 87,131)( 88,133)( 89,135)( 90,134)(137,138)(140,141)(143,144)(146,147)
(149,150)(152,153)(155,156)(158,159)(161,162)(164,165)(167,168)(170,171)
(173,174)(176,177)(179,180)(181,226)(182,228)(183,227)(184,229)(185,231)
(186,230)(187,232)(188,234)(189,233)(190,235)(191,237)(192,236)(193,238)
(194,240)(195,239)(196,241)(197,243)(198,242)(199,244)(200,246)(201,245)
(202,247)(203,249)(204,248)(205,250)(206,252)(207,251)(208,253)(209,255)
(210,254)(211,256)(212,258)(213,257)(214,259)(215,261)(216,260)(217,262)
(218,264)(219,263)(220,265)(221,267)(222,266)(223,268)(224,270)(225,269);;
s1 := ( 1,181)( 2,183)( 3,182)( 4,184)( 5,186)( 6,185)( 7,187)( 8,189)
( 9,188)( 10,190)( 11,192)( 12,191)( 13,193)( 14,195)( 15,194)( 16,197)
( 17,196)( 18,198)( 19,200)( 20,199)( 21,201)( 22,203)( 23,202)( 24,204)
( 25,206)( 26,205)( 27,207)( 28,209)( 29,208)( 30,210)( 31,213)( 32,212)
( 33,211)( 34,216)( 35,215)( 36,214)( 37,219)( 38,218)( 39,217)( 40,222)
( 41,221)( 42,220)( 43,225)( 44,224)( 45,223)( 46,136)( 47,138)( 48,137)
( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)( 56,147)
( 57,146)( 58,148)( 59,150)( 60,149)( 61,152)( 62,151)( 63,153)( 64,155)
( 65,154)( 66,156)( 67,158)( 68,157)( 69,159)( 70,161)( 71,160)( 72,162)
( 73,164)( 74,163)( 75,165)( 76,168)( 77,167)( 78,166)( 79,171)( 80,170)
( 81,169)( 82,174)( 83,173)( 84,172)( 85,177)( 86,176)( 87,175)( 88,180)
( 89,179)( 90,178)( 91,226)( 92,228)( 93,227)( 94,229)( 95,231)( 96,230)
( 97,232)( 98,234)( 99,233)(100,235)(101,237)(102,236)(103,238)(104,240)
(105,239)(106,242)(107,241)(108,243)(109,245)(110,244)(111,246)(112,248)
(113,247)(114,249)(115,251)(116,250)(117,252)(118,254)(119,253)(120,255)
(121,258)(122,257)(123,256)(124,261)(125,260)(126,259)(127,264)(128,263)
(129,262)(130,267)(131,266)(132,265)(133,270)(134,269)(135,268);;
s2 := ( 1, 16)( 2, 18)( 3, 17)( 4, 28)( 5, 30)( 6, 29)( 7, 25)( 8, 27)
( 9, 26)( 10, 22)( 11, 24)( 12, 23)( 13, 19)( 14, 21)( 15, 20)( 32, 33)
( 34, 43)( 35, 45)( 36, 44)( 37, 40)( 38, 42)( 39, 41)( 46, 61)( 47, 63)
( 48, 62)( 49, 73)( 50, 75)( 51, 74)( 52, 70)( 53, 72)( 54, 71)( 55, 67)
( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 77, 78)( 79, 88)( 80, 90)
( 81, 89)( 82, 85)( 83, 87)( 84, 86)( 91,106)( 92,108)( 93,107)( 94,118)
( 95,120)( 96,119)( 97,115)( 98,117)( 99,116)(100,112)(101,114)(102,113)
(103,109)(104,111)(105,110)(122,123)(124,133)(125,135)(126,134)(127,130)
(128,132)(129,131)(136,151)(137,153)(138,152)(139,163)(140,165)(141,164)
(142,160)(143,162)(144,161)(145,157)(146,159)(147,158)(148,154)(149,156)
(150,155)(167,168)(169,178)(170,180)(171,179)(172,175)(173,177)(174,176)
(181,196)(182,198)(183,197)(184,208)(185,210)(186,209)(187,205)(188,207)
(189,206)(190,202)(191,204)(192,203)(193,199)(194,201)(195,200)(212,213)
(214,223)(215,225)(216,224)(217,220)(218,222)(219,221)(226,241)(227,243)
(228,242)(229,253)(230,255)(231,254)(232,250)(233,252)(234,251)(235,247)
(236,249)(237,248)(238,244)(239,246)(240,245)(257,258)(259,268)(260,270)
(261,269)(262,265)(263,267)(264,266);;
s3 := ( 1, 4)( 2, 6)( 3, 5)( 7, 13)( 8, 15)( 9, 14)( 11, 12)( 16, 34)
( 17, 36)( 18, 35)( 19, 31)( 20, 33)( 21, 32)( 22, 43)( 23, 45)( 24, 44)
( 25, 40)( 26, 42)( 27, 41)( 28, 37)( 29, 39)( 30, 38)( 46, 49)( 47, 51)
( 48, 50)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 61, 79)( 62, 81)( 63, 80)
( 64, 76)( 65, 78)( 66, 77)( 67, 88)( 68, 90)( 69, 89)( 70, 85)( 71, 87)
( 72, 86)( 73, 82)( 74, 84)( 75, 83)( 91, 94)( 92, 96)( 93, 95)( 97,103)
( 98,105)( 99,104)(101,102)(106,124)(107,126)(108,125)(109,121)(110,123)
(111,122)(112,133)(113,135)(114,134)(115,130)(116,132)(117,131)(118,127)
(119,129)(120,128)(136,139)(137,141)(138,140)(142,148)(143,150)(144,149)
(146,147)(151,169)(152,171)(153,170)(154,166)(155,168)(156,167)(157,178)
(158,180)(159,179)(160,175)(161,177)(162,176)(163,172)(164,174)(165,173)
(181,184)(182,186)(183,185)(187,193)(188,195)(189,194)(191,192)(196,214)
(197,216)(198,215)(199,211)(200,213)(201,212)(202,223)(203,225)(204,224)
(205,220)(206,222)(207,221)(208,217)(209,219)(210,218)(226,229)(227,231)
(228,230)(232,238)(233,240)(234,239)(236,237)(241,259)(242,261)(243,260)
(244,256)(245,258)(246,257)(247,268)(248,270)(249,269)(250,265)(251,267)
(252,266)(253,262)(254,264)(255,263);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(270)!( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)
( 46, 91)( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)
( 54, 98)( 55,100)( 56,102)( 57,101)( 58,103)( 59,105)( 60,104)( 61,106)
( 62,108)( 63,107)( 64,109)( 65,111)( 66,110)( 67,112)( 68,114)( 69,113)
( 70,115)( 71,117)( 72,116)( 73,118)( 74,120)( 75,119)( 76,121)( 77,123)
( 78,122)( 79,124)( 80,126)( 81,125)( 82,127)( 83,129)( 84,128)( 85,130)
( 86,132)( 87,131)( 88,133)( 89,135)( 90,134)(137,138)(140,141)(143,144)
(146,147)(149,150)(152,153)(155,156)(158,159)(161,162)(164,165)(167,168)
(170,171)(173,174)(176,177)(179,180)(181,226)(182,228)(183,227)(184,229)
(185,231)(186,230)(187,232)(188,234)(189,233)(190,235)(191,237)(192,236)
(193,238)(194,240)(195,239)(196,241)(197,243)(198,242)(199,244)(200,246)
(201,245)(202,247)(203,249)(204,248)(205,250)(206,252)(207,251)(208,253)
(209,255)(210,254)(211,256)(212,258)(213,257)(214,259)(215,261)(216,260)
(217,262)(218,264)(219,263)(220,265)(221,267)(222,266)(223,268)(224,270)
(225,269);
s1 := Sym(270)!( 1,181)( 2,183)( 3,182)( 4,184)( 5,186)( 6,185)( 7,187)
( 8,189)( 9,188)( 10,190)( 11,192)( 12,191)( 13,193)( 14,195)( 15,194)
( 16,197)( 17,196)( 18,198)( 19,200)( 20,199)( 21,201)( 22,203)( 23,202)
( 24,204)( 25,206)( 26,205)( 27,207)( 28,209)( 29,208)( 30,210)( 31,213)
( 32,212)( 33,211)( 34,216)( 35,215)( 36,214)( 37,219)( 38,218)( 39,217)
( 40,222)( 41,221)( 42,220)( 43,225)( 44,224)( 45,223)( 46,136)( 47,138)
( 48,137)( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)
( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,152)( 62,151)( 63,153)
( 64,155)( 65,154)( 66,156)( 67,158)( 68,157)( 69,159)( 70,161)( 71,160)
( 72,162)( 73,164)( 74,163)( 75,165)( 76,168)( 77,167)( 78,166)( 79,171)
( 80,170)( 81,169)( 82,174)( 83,173)( 84,172)( 85,177)( 86,176)( 87,175)
( 88,180)( 89,179)( 90,178)( 91,226)( 92,228)( 93,227)( 94,229)( 95,231)
( 96,230)( 97,232)( 98,234)( 99,233)(100,235)(101,237)(102,236)(103,238)
(104,240)(105,239)(106,242)(107,241)(108,243)(109,245)(110,244)(111,246)
(112,248)(113,247)(114,249)(115,251)(116,250)(117,252)(118,254)(119,253)
(120,255)(121,258)(122,257)(123,256)(124,261)(125,260)(126,259)(127,264)
(128,263)(129,262)(130,267)(131,266)(132,265)(133,270)(134,269)(135,268);
s2 := Sym(270)!( 1, 16)( 2, 18)( 3, 17)( 4, 28)( 5, 30)( 6, 29)( 7, 25)
( 8, 27)( 9, 26)( 10, 22)( 11, 24)( 12, 23)( 13, 19)( 14, 21)( 15, 20)
( 32, 33)( 34, 43)( 35, 45)( 36, 44)( 37, 40)( 38, 42)( 39, 41)( 46, 61)
( 47, 63)( 48, 62)( 49, 73)( 50, 75)( 51, 74)( 52, 70)( 53, 72)( 54, 71)
( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 77, 78)( 79, 88)
( 80, 90)( 81, 89)( 82, 85)( 83, 87)( 84, 86)( 91,106)( 92,108)( 93,107)
( 94,118)( 95,120)( 96,119)( 97,115)( 98,117)( 99,116)(100,112)(101,114)
(102,113)(103,109)(104,111)(105,110)(122,123)(124,133)(125,135)(126,134)
(127,130)(128,132)(129,131)(136,151)(137,153)(138,152)(139,163)(140,165)
(141,164)(142,160)(143,162)(144,161)(145,157)(146,159)(147,158)(148,154)
(149,156)(150,155)(167,168)(169,178)(170,180)(171,179)(172,175)(173,177)
(174,176)(181,196)(182,198)(183,197)(184,208)(185,210)(186,209)(187,205)
(188,207)(189,206)(190,202)(191,204)(192,203)(193,199)(194,201)(195,200)
(212,213)(214,223)(215,225)(216,224)(217,220)(218,222)(219,221)(226,241)
(227,243)(228,242)(229,253)(230,255)(231,254)(232,250)(233,252)(234,251)
(235,247)(236,249)(237,248)(238,244)(239,246)(240,245)(257,258)(259,268)
(260,270)(261,269)(262,265)(263,267)(264,266);
s3 := Sym(270)!( 1, 4)( 2, 6)( 3, 5)( 7, 13)( 8, 15)( 9, 14)( 11, 12)
( 16, 34)( 17, 36)( 18, 35)( 19, 31)( 20, 33)( 21, 32)( 22, 43)( 23, 45)
( 24, 44)( 25, 40)( 26, 42)( 27, 41)( 28, 37)( 29, 39)( 30, 38)( 46, 49)
( 47, 51)( 48, 50)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 61, 79)( 62, 81)
( 63, 80)( 64, 76)( 65, 78)( 66, 77)( 67, 88)( 68, 90)( 69, 89)( 70, 85)
( 71, 87)( 72, 86)( 73, 82)( 74, 84)( 75, 83)( 91, 94)( 92, 96)( 93, 95)
( 97,103)( 98,105)( 99,104)(101,102)(106,124)(107,126)(108,125)(109,121)
(110,123)(111,122)(112,133)(113,135)(114,134)(115,130)(116,132)(117,131)
(118,127)(119,129)(120,128)(136,139)(137,141)(138,140)(142,148)(143,150)
(144,149)(146,147)(151,169)(152,171)(153,170)(154,166)(155,168)(156,167)
(157,178)(158,180)(159,179)(160,175)(161,177)(162,176)(163,172)(164,174)
(165,173)(181,184)(182,186)(183,185)(187,193)(188,195)(189,194)(191,192)
(196,214)(197,216)(198,215)(199,211)(200,213)(201,212)(202,223)(203,225)
(204,224)(205,220)(206,222)(207,221)(208,217)(209,219)(210,218)(226,229)
(227,231)(228,230)(232,238)(233,240)(234,239)(236,237)(241,259)(242,261)
(243,260)(244,256)(245,258)(246,257)(247,268)(248,270)(249,269)(250,265)
(251,267)(252,266)(253,262)(254,264)(255,263);
poly := sub<Sym(270)|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,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
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 >;
References : None.
to this polytope