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