Polytope of Type {12,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*1728p
if this polytope has a name.
Group : SmallGroup(1728,21980)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 36, 432, 72
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {12,12}*864h
3-fold quotients : {12,24}*576a, {12,24}*576e, {12,24}*576f
4-fold quotients : {6,12}*432g, {12,6}*432g
6-fold quotients : {12,12}*288a, {12,12}*288b, {12,12}*288c
8-fold quotients : {6,6}*216d
9-fold quotients : {4,24}*192b, {12,8}*192b
12-fold quotients : {6,12}*144a, {6,12}*144b, {12,6}*144a, {12,6}*144b, {6,12}*144c, {12,6}*144c
18-fold quotients : {4,12}*96a, {12,4}*96a
24-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
27-fold quotients : {4,8}*64b
36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
48-fold quotients : {3,6}*36, {6,3}*36
54-fold quotients : {4,4}*32
72-fold quotients : {2,6}*24, {6,2}*24
108-fold quotients : {2,4}*16, {4,2}*16
144-fold quotients : {2,3}*12, {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,325)( 2,326)( 3,327)( 4,331)( 5,332)( 6,333)( 7,328)( 8,329)
( 9,330)( 10,343)( 11,344)( 12,345)( 13,349)( 14,350)( 15,351)( 16,346)
( 17,347)( 18,348)( 19,334)( 20,335)( 21,336)( 22,340)( 23,341)( 24,342)
( 25,337)( 26,338)( 27,339)( 28,352)( 29,353)( 30,354)( 31,358)( 32,359)
( 33,360)( 34,355)( 35,356)( 36,357)( 37,370)( 38,371)( 39,372)( 40,376)
( 41,377)( 42,378)( 43,373)( 44,374)( 45,375)( 46,361)( 47,362)( 48,363)
( 49,367)( 50,368)( 51,369)( 52,364)( 53,365)( 54,366)( 55,406)( 56,407)
( 57,408)( 58,412)( 59,413)( 60,414)( 61,409)( 62,410)( 63,411)( 64,424)
( 65,425)( 66,426)( 67,430)( 68,431)( 69,432)( 70,427)( 71,428)( 72,429)
( 73,415)( 74,416)( 75,417)( 76,421)( 77,422)( 78,423)( 79,418)( 80,419)
( 81,420)( 82,379)( 83,380)( 84,381)( 85,385)( 86,386)( 87,387)( 88,382)
( 89,383)( 90,384)( 91,397)( 92,398)( 93,399)( 94,403)( 95,404)( 96,405)
( 97,400)( 98,401)( 99,402)(100,388)(101,389)(102,390)(103,394)(104,395)
(105,396)(106,391)(107,392)(108,393)(109,271)(110,272)(111,273)(112,277)
(113,278)(114,279)(115,274)(116,275)(117,276)(118,289)(119,290)(120,291)
(121,295)(122,296)(123,297)(124,292)(125,293)(126,294)(127,280)(128,281)
(129,282)(130,286)(131,287)(132,288)(133,283)(134,284)(135,285)(136,298)
(137,299)(138,300)(139,304)(140,305)(141,306)(142,301)(143,302)(144,303)
(145,316)(146,317)(147,318)(148,322)(149,323)(150,324)(151,319)(152,320)
(153,321)(154,307)(155,308)(156,309)(157,313)(158,314)(159,315)(160,310)
(161,311)(162,312)(163,217)(164,218)(165,219)(166,223)(167,224)(168,225)
(169,220)(170,221)(171,222)(172,235)(173,236)(174,237)(175,241)(176,242)
(177,243)(178,238)(179,239)(180,240)(181,226)(182,227)(183,228)(184,232)
(185,233)(186,234)(187,229)(188,230)(189,231)(190,244)(191,245)(192,246)
(193,250)(194,251)(195,252)(196,247)(197,248)(198,249)(199,262)(200,263)
(201,264)(202,268)(203,269)(204,270)(205,265)(206,266)(207,267)(208,253)
(209,254)(210,255)(211,259)(212,260)(213,261)(214,256)(215,257)(216,258);;
s1 := ( 1,337)( 2,339)( 3,338)( 4,334)( 5,336)( 6,335)( 7,340)( 8,342)
( 9,341)( 10,328)( 11,330)( 12,329)( 13,325)( 14,327)( 15,326)( 16,331)
( 17,333)( 18,332)( 19,346)( 20,348)( 21,347)( 22,343)( 23,345)( 24,344)
( 25,349)( 26,351)( 27,350)( 28,364)( 29,366)( 30,365)( 31,361)( 32,363)
( 33,362)( 34,367)( 35,369)( 36,368)( 37,355)( 38,357)( 39,356)( 40,352)
( 41,354)( 42,353)( 43,358)( 44,360)( 45,359)( 46,373)( 47,375)( 48,374)
( 49,370)( 50,372)( 51,371)( 52,376)( 53,378)( 54,377)( 55,418)( 56,420)
( 57,419)( 58,415)( 59,417)( 60,416)( 61,421)( 62,423)( 63,422)( 64,409)
( 65,411)( 66,410)( 67,406)( 68,408)( 69,407)( 70,412)( 71,414)( 72,413)
( 73,427)( 74,429)( 75,428)( 76,424)( 77,426)( 78,425)( 79,430)( 80,432)
( 81,431)( 82,391)( 83,393)( 84,392)( 85,388)( 86,390)( 87,389)( 88,394)
( 89,396)( 90,395)( 91,382)( 92,384)( 93,383)( 94,379)( 95,381)( 96,380)
( 97,385)( 98,387)( 99,386)(100,400)(101,402)(102,401)(103,397)(104,399)
(105,398)(106,403)(107,405)(108,404)(109,229)(110,231)(111,230)(112,226)
(113,228)(114,227)(115,232)(116,234)(117,233)(118,220)(119,222)(120,221)
(121,217)(122,219)(123,218)(124,223)(125,225)(126,224)(127,238)(128,240)
(129,239)(130,235)(131,237)(132,236)(133,241)(134,243)(135,242)(136,256)
(137,258)(138,257)(139,253)(140,255)(141,254)(142,259)(143,261)(144,260)
(145,247)(146,249)(147,248)(148,244)(149,246)(150,245)(151,250)(152,252)
(153,251)(154,265)(155,267)(156,266)(157,262)(158,264)(159,263)(160,268)
(161,270)(162,269)(163,310)(164,312)(165,311)(166,307)(167,309)(168,308)
(169,313)(170,315)(171,314)(172,301)(173,303)(174,302)(175,298)(176,300)
(177,299)(178,304)(179,306)(180,305)(181,319)(182,321)(183,320)(184,316)
(185,318)(186,317)(187,322)(188,324)(189,323)(190,283)(191,285)(192,284)
(193,280)(194,282)(195,281)(196,286)(197,288)(198,287)(199,274)(200,276)
(201,275)(202,271)(203,273)(204,272)(205,277)(206,279)(207,278)(208,292)
(209,294)(210,293)(211,289)(212,291)(213,290)(214,295)(215,297)(216,296);;
s2 := ( 1,272)( 2,271)( 3,273)( 4,275)( 5,274)( 6,276)( 7,278)( 8,277)
( 9,279)( 10,290)( 11,289)( 12,291)( 13,293)( 14,292)( 15,294)( 16,296)
( 17,295)( 18,297)( 19,281)( 20,280)( 21,282)( 22,284)( 23,283)( 24,285)
( 25,287)( 26,286)( 27,288)( 28,299)( 29,298)( 30,300)( 31,302)( 32,301)
( 33,303)( 34,305)( 35,304)( 36,306)( 37,317)( 38,316)( 39,318)( 40,320)
( 41,319)( 42,321)( 43,323)( 44,322)( 45,324)( 46,308)( 47,307)( 48,309)
( 49,311)( 50,310)( 51,312)( 52,314)( 53,313)( 54,315)( 55,218)( 56,217)
( 57,219)( 58,221)( 59,220)( 60,222)( 61,224)( 62,223)( 63,225)( 64,236)
( 65,235)( 66,237)( 67,239)( 68,238)( 69,240)( 70,242)( 71,241)( 72,243)
( 73,227)( 74,226)( 75,228)( 76,230)( 77,229)( 78,231)( 79,233)( 80,232)
( 81,234)( 82,245)( 83,244)( 84,246)( 85,248)( 86,247)( 87,249)( 88,251)
( 89,250)( 90,252)( 91,263)( 92,262)( 93,264)( 94,266)( 95,265)( 96,267)
( 97,269)( 98,268)( 99,270)(100,254)(101,253)(102,255)(103,257)(104,256)
(105,258)(106,260)(107,259)(108,261)(109,326)(110,325)(111,327)(112,329)
(113,328)(114,330)(115,332)(116,331)(117,333)(118,344)(119,343)(120,345)
(121,347)(122,346)(123,348)(124,350)(125,349)(126,351)(127,335)(128,334)
(129,336)(130,338)(131,337)(132,339)(133,341)(134,340)(135,342)(136,353)
(137,352)(138,354)(139,356)(140,355)(141,357)(142,359)(143,358)(144,360)
(145,371)(146,370)(147,372)(148,374)(149,373)(150,375)(151,377)(152,376)
(153,378)(154,362)(155,361)(156,363)(157,365)(158,364)(159,366)(160,368)
(161,367)(162,369)(163,407)(164,406)(165,408)(166,410)(167,409)(168,411)
(169,413)(170,412)(171,414)(172,425)(173,424)(174,426)(175,428)(176,427)
(177,429)(178,431)(179,430)(180,432)(181,416)(182,415)(183,417)(184,419)
(185,418)(186,420)(187,422)(188,421)(189,423)(190,380)(191,379)(192,381)
(193,383)(194,382)(195,384)(196,386)(197,385)(198,387)(199,398)(200,397)
(201,399)(202,401)(203,400)(204,402)(205,404)(206,403)(207,405)(208,389)
(209,388)(210,390)(211,392)(212,391)(213,393)(214,395)(215,394)(216,396);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*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,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 1,325)( 2,326)( 3,327)( 4,331)( 5,332)( 6,333)( 7,328)
( 8,329)( 9,330)( 10,343)( 11,344)( 12,345)( 13,349)( 14,350)( 15,351)
( 16,346)( 17,347)( 18,348)( 19,334)( 20,335)( 21,336)( 22,340)( 23,341)
( 24,342)( 25,337)( 26,338)( 27,339)( 28,352)( 29,353)( 30,354)( 31,358)
( 32,359)( 33,360)( 34,355)( 35,356)( 36,357)( 37,370)( 38,371)( 39,372)
( 40,376)( 41,377)( 42,378)( 43,373)( 44,374)( 45,375)( 46,361)( 47,362)
( 48,363)( 49,367)( 50,368)( 51,369)( 52,364)( 53,365)( 54,366)( 55,406)
( 56,407)( 57,408)( 58,412)( 59,413)( 60,414)( 61,409)( 62,410)( 63,411)
( 64,424)( 65,425)( 66,426)( 67,430)( 68,431)( 69,432)( 70,427)( 71,428)
( 72,429)( 73,415)( 74,416)( 75,417)( 76,421)( 77,422)( 78,423)( 79,418)
( 80,419)( 81,420)( 82,379)( 83,380)( 84,381)( 85,385)( 86,386)( 87,387)
( 88,382)( 89,383)( 90,384)( 91,397)( 92,398)( 93,399)( 94,403)( 95,404)
( 96,405)( 97,400)( 98,401)( 99,402)(100,388)(101,389)(102,390)(103,394)
(104,395)(105,396)(106,391)(107,392)(108,393)(109,271)(110,272)(111,273)
(112,277)(113,278)(114,279)(115,274)(116,275)(117,276)(118,289)(119,290)
(120,291)(121,295)(122,296)(123,297)(124,292)(125,293)(126,294)(127,280)
(128,281)(129,282)(130,286)(131,287)(132,288)(133,283)(134,284)(135,285)
(136,298)(137,299)(138,300)(139,304)(140,305)(141,306)(142,301)(143,302)
(144,303)(145,316)(146,317)(147,318)(148,322)(149,323)(150,324)(151,319)
(152,320)(153,321)(154,307)(155,308)(156,309)(157,313)(158,314)(159,315)
(160,310)(161,311)(162,312)(163,217)(164,218)(165,219)(166,223)(167,224)
(168,225)(169,220)(170,221)(171,222)(172,235)(173,236)(174,237)(175,241)
(176,242)(177,243)(178,238)(179,239)(180,240)(181,226)(182,227)(183,228)
(184,232)(185,233)(186,234)(187,229)(188,230)(189,231)(190,244)(191,245)
(192,246)(193,250)(194,251)(195,252)(196,247)(197,248)(198,249)(199,262)
(200,263)(201,264)(202,268)(203,269)(204,270)(205,265)(206,266)(207,267)
(208,253)(209,254)(210,255)(211,259)(212,260)(213,261)(214,256)(215,257)
(216,258);
s1 := Sym(432)!( 1,337)( 2,339)( 3,338)( 4,334)( 5,336)( 6,335)( 7,340)
( 8,342)( 9,341)( 10,328)( 11,330)( 12,329)( 13,325)( 14,327)( 15,326)
( 16,331)( 17,333)( 18,332)( 19,346)( 20,348)( 21,347)( 22,343)( 23,345)
( 24,344)( 25,349)( 26,351)( 27,350)( 28,364)( 29,366)( 30,365)( 31,361)
( 32,363)( 33,362)( 34,367)( 35,369)( 36,368)( 37,355)( 38,357)( 39,356)
( 40,352)( 41,354)( 42,353)( 43,358)( 44,360)( 45,359)( 46,373)( 47,375)
( 48,374)( 49,370)( 50,372)( 51,371)( 52,376)( 53,378)( 54,377)( 55,418)
( 56,420)( 57,419)( 58,415)( 59,417)( 60,416)( 61,421)( 62,423)( 63,422)
( 64,409)( 65,411)( 66,410)( 67,406)( 68,408)( 69,407)( 70,412)( 71,414)
( 72,413)( 73,427)( 74,429)( 75,428)( 76,424)( 77,426)( 78,425)( 79,430)
( 80,432)( 81,431)( 82,391)( 83,393)( 84,392)( 85,388)( 86,390)( 87,389)
( 88,394)( 89,396)( 90,395)( 91,382)( 92,384)( 93,383)( 94,379)( 95,381)
( 96,380)( 97,385)( 98,387)( 99,386)(100,400)(101,402)(102,401)(103,397)
(104,399)(105,398)(106,403)(107,405)(108,404)(109,229)(110,231)(111,230)
(112,226)(113,228)(114,227)(115,232)(116,234)(117,233)(118,220)(119,222)
(120,221)(121,217)(122,219)(123,218)(124,223)(125,225)(126,224)(127,238)
(128,240)(129,239)(130,235)(131,237)(132,236)(133,241)(134,243)(135,242)
(136,256)(137,258)(138,257)(139,253)(140,255)(141,254)(142,259)(143,261)
(144,260)(145,247)(146,249)(147,248)(148,244)(149,246)(150,245)(151,250)
(152,252)(153,251)(154,265)(155,267)(156,266)(157,262)(158,264)(159,263)
(160,268)(161,270)(162,269)(163,310)(164,312)(165,311)(166,307)(167,309)
(168,308)(169,313)(170,315)(171,314)(172,301)(173,303)(174,302)(175,298)
(176,300)(177,299)(178,304)(179,306)(180,305)(181,319)(182,321)(183,320)
(184,316)(185,318)(186,317)(187,322)(188,324)(189,323)(190,283)(191,285)
(192,284)(193,280)(194,282)(195,281)(196,286)(197,288)(198,287)(199,274)
(200,276)(201,275)(202,271)(203,273)(204,272)(205,277)(206,279)(207,278)
(208,292)(209,294)(210,293)(211,289)(212,291)(213,290)(214,295)(215,297)
(216,296);
s2 := Sym(432)!( 1,272)( 2,271)( 3,273)( 4,275)( 5,274)( 6,276)( 7,278)
( 8,277)( 9,279)( 10,290)( 11,289)( 12,291)( 13,293)( 14,292)( 15,294)
( 16,296)( 17,295)( 18,297)( 19,281)( 20,280)( 21,282)( 22,284)( 23,283)
( 24,285)( 25,287)( 26,286)( 27,288)( 28,299)( 29,298)( 30,300)( 31,302)
( 32,301)( 33,303)( 34,305)( 35,304)( 36,306)( 37,317)( 38,316)( 39,318)
( 40,320)( 41,319)( 42,321)( 43,323)( 44,322)( 45,324)( 46,308)( 47,307)
( 48,309)( 49,311)( 50,310)( 51,312)( 52,314)( 53,313)( 54,315)( 55,218)
( 56,217)( 57,219)( 58,221)( 59,220)( 60,222)( 61,224)( 62,223)( 63,225)
( 64,236)( 65,235)( 66,237)( 67,239)( 68,238)( 69,240)( 70,242)( 71,241)
( 72,243)( 73,227)( 74,226)( 75,228)( 76,230)( 77,229)( 78,231)( 79,233)
( 80,232)( 81,234)( 82,245)( 83,244)( 84,246)( 85,248)( 86,247)( 87,249)
( 88,251)( 89,250)( 90,252)( 91,263)( 92,262)( 93,264)( 94,266)( 95,265)
( 96,267)( 97,269)( 98,268)( 99,270)(100,254)(101,253)(102,255)(103,257)
(104,256)(105,258)(106,260)(107,259)(108,261)(109,326)(110,325)(111,327)
(112,329)(113,328)(114,330)(115,332)(116,331)(117,333)(118,344)(119,343)
(120,345)(121,347)(122,346)(123,348)(124,350)(125,349)(126,351)(127,335)
(128,334)(129,336)(130,338)(131,337)(132,339)(133,341)(134,340)(135,342)
(136,353)(137,352)(138,354)(139,356)(140,355)(141,357)(142,359)(143,358)
(144,360)(145,371)(146,370)(147,372)(148,374)(149,373)(150,375)(151,377)
(152,376)(153,378)(154,362)(155,361)(156,363)(157,365)(158,364)(159,366)
(160,368)(161,367)(162,369)(163,407)(164,406)(165,408)(166,410)(167,409)
(168,411)(169,413)(170,412)(171,414)(172,425)(173,424)(174,426)(175,428)
(176,427)(177,429)(178,431)(179,430)(180,432)(181,416)(182,415)(183,417)
(184,419)(185,418)(186,420)(187,422)(188,421)(189,423)(190,380)(191,379)
(192,381)(193,383)(194,382)(195,384)(196,386)(197,385)(198,387)(199,398)
(200,397)(201,399)(202,401)(203,400)(204,402)(205,404)(206,403)(207,405)
(208,389)(209,388)(210,390)(211,392)(212,391)(213,393)(214,395)(215,394)
(216,396);
poly := sub<Sym(432)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*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,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 >;
References : None.
to this polytope