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