Polytope of Type {3,2,3,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,3,12}*1728
if this polytope has a name.
Group : SmallGroup(1728,46303)
Rank : 5
Schlafli Type : {3,2,3,12}
Number of vertices, edges, etc : 3, 3, 12, 72, 48
Order of s0s1s2s3s4 : 24
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {3,2,3,6}*864
3-fold quotients : {3,2,3,12}*576
6-fold quotients : {3,2,3,6}*288
8-fold quotients : {3,2,3,6}*216
12-fold quotients : {3,2,3,3}*144
24-fold quotients : {3,2,3,2}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4,220)( 5,221)( 6,224)( 7,225)( 8,222)( 9,223)( 10,227)( 11,226)
( 12,236)( 13,237)( 14,240)( 15,241)( 16,238)( 17,239)( 18,243)( 19,242)
( 20,228)( 21,229)( 22,232)( 23,233)( 24,230)( 25,231)( 26,235)( 27,234)
( 28,268)( 29,269)( 30,272)( 31,273)( 32,270)( 33,271)( 34,275)( 35,274)
( 36,284)( 37,285)( 38,288)( 39,289)( 40,286)( 41,287)( 42,291)( 43,290)
( 44,276)( 45,277)( 46,280)( 47,281)( 48,278)( 49,279)( 50,283)( 51,282)
( 52,244)( 53,245)( 54,248)( 55,249)( 56,246)( 57,247)( 58,251)( 59,250)
( 60,260)( 61,261)( 62,264)( 63,265)( 64,262)( 65,263)( 66,267)( 67,266)
( 68,252)( 69,253)( 70,256)( 71,257)( 72,254)( 73,255)( 74,259)( 75,258)
( 76,149)( 77,148)( 78,153)( 79,152)( 80,151)( 81,150)( 82,154)( 83,155)
( 84,165)( 85,164)( 86,169)( 87,168)( 88,167)( 89,166)( 90,170)( 91,171)
( 92,157)( 93,156)( 94,161)( 95,160)( 96,159)( 97,158)( 98,162)( 99,163)
(100,197)(101,196)(102,201)(103,200)(104,199)(105,198)(106,202)(107,203)
(108,213)(109,212)(110,217)(111,216)(112,215)(113,214)(114,218)(115,219)
(116,205)(117,204)(118,209)(119,208)(120,207)(121,206)(122,210)(123,211)
(124,173)(125,172)(126,177)(127,176)(128,175)(129,174)(130,178)(131,179)
(132,189)(133,188)(134,193)(135,192)(136,191)(137,190)(138,194)(139,195)
(140,181)(141,180)(142,185)(143,184)(144,183)(145,182)(146,186)(147,187);;
s3 := ( 4,252)( 5,253)( 6,255)( 7,254)( 8,258)( 9,259)( 10,256)( 11,257)
( 12,244)( 13,245)( 14,247)( 15,246)( 16,250)( 17,251)( 18,248)( 19,249)
( 20,260)( 21,261)( 22,263)( 23,262)( 24,266)( 25,267)( 26,264)( 27,265)
( 28,228)( 29,229)( 30,231)( 31,230)( 32,234)( 33,235)( 34,232)( 35,233)
( 36,220)( 37,221)( 38,223)( 39,222)( 40,226)( 41,227)( 42,224)( 43,225)
( 44,236)( 45,237)( 46,239)( 47,238)( 48,242)( 49,243)( 50,240)( 51,241)
( 52,276)( 53,277)( 54,279)( 55,278)( 56,282)( 57,283)( 58,280)( 59,281)
( 60,268)( 61,269)( 62,271)( 63,270)( 64,274)( 65,275)( 66,272)( 67,273)
( 68,284)( 69,285)( 70,287)( 71,286)( 72,290)( 73,291)( 74,288)( 75,289)
( 76,181)( 77,180)( 78,182)( 79,183)( 80,187)( 81,186)( 82,185)( 83,184)
( 84,173)( 85,172)( 86,174)( 87,175)( 88,179)( 89,178)( 90,177)( 91,176)
( 92,189)( 93,188)( 94,190)( 95,191)( 96,195)( 97,194)( 98,193)( 99,192)
(100,157)(101,156)(102,158)(103,159)(104,163)(105,162)(106,161)(107,160)
(108,149)(109,148)(110,150)(111,151)(112,155)(113,154)(114,153)(115,152)
(116,165)(117,164)(118,166)(119,167)(120,171)(121,170)(122,169)(123,168)
(124,205)(125,204)(126,206)(127,207)(128,211)(129,210)(130,209)(131,208)
(132,197)(133,196)(134,198)(135,199)(136,203)(137,202)(138,201)(139,200)
(140,213)(141,212)(142,214)(143,215)(144,219)(145,218)(146,217)(147,216);;
s4 := ( 4, 82)( 5, 83)( 6, 79)( 7, 78)( 8, 80)( 9, 81)( 10, 76)( 11, 77)
( 12, 90)( 13, 91)( 14, 87)( 15, 86)( 16, 88)( 17, 89)( 18, 84)( 19, 85)
( 20, 98)( 21, 99)( 22, 95)( 23, 94)( 24, 96)( 25, 97)( 26, 92)( 27, 93)
( 28,130)( 29,131)( 30,127)( 31,126)( 32,128)( 33,129)( 34,124)( 35,125)
( 36,138)( 37,139)( 38,135)( 39,134)( 40,136)( 41,137)( 42,132)( 43,133)
( 44,146)( 45,147)( 46,143)( 47,142)( 48,144)( 49,145)( 50,140)( 51,141)
( 52,106)( 53,107)( 54,103)( 55,102)( 56,104)( 57,105)( 58,100)( 59,101)
( 60,114)( 61,115)( 62,111)( 63,110)( 64,112)( 65,113)( 66,108)( 67,109)
( 68,122)( 69,123)( 70,119)( 71,118)( 72,120)( 73,121)( 74,116)( 75,117)
(148,226)(149,227)(150,223)(151,222)(152,224)(153,225)(154,220)(155,221)
(156,234)(157,235)(158,231)(159,230)(160,232)(161,233)(162,228)(163,229)
(164,242)(165,243)(166,239)(167,238)(168,240)(169,241)(170,236)(171,237)
(172,274)(173,275)(174,271)(175,270)(176,272)(177,273)(178,268)(179,269)
(180,282)(181,283)(182,279)(183,278)(184,280)(185,281)(186,276)(187,277)
(188,290)(189,291)(190,287)(191,286)(192,288)(193,289)(194,284)(195,285)
(196,250)(197,251)(198,247)(199,246)(200,248)(201,249)(202,244)(203,245)
(204,258)(205,259)(206,255)(207,254)(208,256)(209,257)(210,252)(211,253)
(212,266)(213,267)(214,263)(215,262)(216,264)(217,265)(218,260)(219,261);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(291)!(2,3);
s1 := Sym(291)!(1,2);
s2 := Sym(291)!( 4,220)( 5,221)( 6,224)( 7,225)( 8,222)( 9,223)( 10,227)
( 11,226)( 12,236)( 13,237)( 14,240)( 15,241)( 16,238)( 17,239)( 18,243)
( 19,242)( 20,228)( 21,229)( 22,232)( 23,233)( 24,230)( 25,231)( 26,235)
( 27,234)( 28,268)( 29,269)( 30,272)( 31,273)( 32,270)( 33,271)( 34,275)
( 35,274)( 36,284)( 37,285)( 38,288)( 39,289)( 40,286)( 41,287)( 42,291)
( 43,290)( 44,276)( 45,277)( 46,280)( 47,281)( 48,278)( 49,279)( 50,283)
( 51,282)( 52,244)( 53,245)( 54,248)( 55,249)( 56,246)( 57,247)( 58,251)
( 59,250)( 60,260)( 61,261)( 62,264)( 63,265)( 64,262)( 65,263)( 66,267)
( 67,266)( 68,252)( 69,253)( 70,256)( 71,257)( 72,254)( 73,255)( 74,259)
( 75,258)( 76,149)( 77,148)( 78,153)( 79,152)( 80,151)( 81,150)( 82,154)
( 83,155)( 84,165)( 85,164)( 86,169)( 87,168)( 88,167)( 89,166)( 90,170)
( 91,171)( 92,157)( 93,156)( 94,161)( 95,160)( 96,159)( 97,158)( 98,162)
( 99,163)(100,197)(101,196)(102,201)(103,200)(104,199)(105,198)(106,202)
(107,203)(108,213)(109,212)(110,217)(111,216)(112,215)(113,214)(114,218)
(115,219)(116,205)(117,204)(118,209)(119,208)(120,207)(121,206)(122,210)
(123,211)(124,173)(125,172)(126,177)(127,176)(128,175)(129,174)(130,178)
(131,179)(132,189)(133,188)(134,193)(135,192)(136,191)(137,190)(138,194)
(139,195)(140,181)(141,180)(142,185)(143,184)(144,183)(145,182)(146,186)
(147,187);
s3 := Sym(291)!( 4,252)( 5,253)( 6,255)( 7,254)( 8,258)( 9,259)( 10,256)
( 11,257)( 12,244)( 13,245)( 14,247)( 15,246)( 16,250)( 17,251)( 18,248)
( 19,249)( 20,260)( 21,261)( 22,263)( 23,262)( 24,266)( 25,267)( 26,264)
( 27,265)( 28,228)( 29,229)( 30,231)( 31,230)( 32,234)( 33,235)( 34,232)
( 35,233)( 36,220)( 37,221)( 38,223)( 39,222)( 40,226)( 41,227)( 42,224)
( 43,225)( 44,236)( 45,237)( 46,239)( 47,238)( 48,242)( 49,243)( 50,240)
( 51,241)( 52,276)( 53,277)( 54,279)( 55,278)( 56,282)( 57,283)( 58,280)
( 59,281)( 60,268)( 61,269)( 62,271)( 63,270)( 64,274)( 65,275)( 66,272)
( 67,273)( 68,284)( 69,285)( 70,287)( 71,286)( 72,290)( 73,291)( 74,288)
( 75,289)( 76,181)( 77,180)( 78,182)( 79,183)( 80,187)( 81,186)( 82,185)
( 83,184)( 84,173)( 85,172)( 86,174)( 87,175)( 88,179)( 89,178)( 90,177)
( 91,176)( 92,189)( 93,188)( 94,190)( 95,191)( 96,195)( 97,194)( 98,193)
( 99,192)(100,157)(101,156)(102,158)(103,159)(104,163)(105,162)(106,161)
(107,160)(108,149)(109,148)(110,150)(111,151)(112,155)(113,154)(114,153)
(115,152)(116,165)(117,164)(118,166)(119,167)(120,171)(121,170)(122,169)
(123,168)(124,205)(125,204)(126,206)(127,207)(128,211)(129,210)(130,209)
(131,208)(132,197)(133,196)(134,198)(135,199)(136,203)(137,202)(138,201)
(139,200)(140,213)(141,212)(142,214)(143,215)(144,219)(145,218)(146,217)
(147,216);
s4 := Sym(291)!( 4, 82)( 5, 83)( 6, 79)( 7, 78)( 8, 80)( 9, 81)( 10, 76)
( 11, 77)( 12, 90)( 13, 91)( 14, 87)( 15, 86)( 16, 88)( 17, 89)( 18, 84)
( 19, 85)( 20, 98)( 21, 99)( 22, 95)( 23, 94)( 24, 96)( 25, 97)( 26, 92)
( 27, 93)( 28,130)( 29,131)( 30,127)( 31,126)( 32,128)( 33,129)( 34,124)
( 35,125)( 36,138)( 37,139)( 38,135)( 39,134)( 40,136)( 41,137)( 42,132)
( 43,133)( 44,146)( 45,147)( 46,143)( 47,142)( 48,144)( 49,145)( 50,140)
( 51,141)( 52,106)( 53,107)( 54,103)( 55,102)( 56,104)( 57,105)( 58,100)
( 59,101)( 60,114)( 61,115)( 62,111)( 63,110)( 64,112)( 65,113)( 66,108)
( 67,109)( 68,122)( 69,123)( 70,119)( 71,118)( 72,120)( 73,121)( 74,116)
( 75,117)(148,226)(149,227)(150,223)(151,222)(152,224)(153,225)(154,220)
(155,221)(156,234)(157,235)(158,231)(159,230)(160,232)(161,233)(162,228)
(163,229)(164,242)(165,243)(166,239)(167,238)(168,240)(169,241)(170,236)
(171,237)(172,274)(173,275)(174,271)(175,270)(176,272)(177,273)(178,268)
(179,269)(180,282)(181,283)(182,279)(183,278)(184,280)(185,281)(186,276)
(187,277)(188,290)(189,291)(190,287)(191,286)(192,288)(193,289)(194,284)
(195,285)(196,250)(197,251)(198,247)(199,246)(200,248)(201,249)(202,244)
(203,245)(204,258)(205,259)(206,255)(207,254)(208,256)(209,257)(210,252)
(211,253)(212,266)(213,267)(214,263)(215,262)(216,264)(217,265)(218,260)
(219,261);
poly := sub<Sym(291)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s2*s3 >;
to this polytope