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# Polytope of Type {24,6}

Atlas Canonical Name : {24,6}*864f
if this polytope has a name.
Group : SmallGroup(864,2800)
Rank : 3
Schlafli Type : {24,6}
Number of vertices, edges, etc : 72, 216, 18
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{24,6,2} of size 1728
Vertex Figure Of :
{2,24,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,6}*432g
3-fold quotients : {24,6}*288a, {24,6}*288b, {24,6}*288c
4-fold quotients : {6,6}*216d
6-fold quotients : {12,6}*144a, {12,6}*144b, {12,6}*144c
9-fold quotients : {24,2}*96, {8,6}*96
12-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
18-fold quotients : {12,2}*48, {4,6}*48a
24-fold quotients : {3,6}*36, {6,3}*36
27-fold quotients : {8,2}*32
36-fold quotients : {2,6}*24, {6,2}*24
54-fold quotients : {4,2}*16
72-fold quotients : {2,3}*12, {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {48,6}*1728f, {24,12}*1728o
Permutation Representation (GAP) :
```s0 := (  1,217)(  2,219)(  3,218)(  4,223)(  5,225)(  6,224)(  7,220)(  8,222)
(  9,221)( 10,226)( 11,228)( 12,227)( 13,232)( 14,234)( 15,233)( 16,229)
( 17,231)( 18,230)( 19,235)( 20,237)( 21,236)( 22,241)( 23,243)( 24,242)
( 25,238)( 26,240)( 27,239)( 28,244)( 29,246)( 30,245)( 31,250)( 32,252)
( 33,251)( 34,247)( 35,249)( 36,248)( 37,253)( 38,255)( 39,254)( 40,259)
( 41,261)( 42,260)( 43,256)( 44,258)( 45,257)( 46,262)( 47,264)( 48,263)
( 49,268)( 50,270)( 51,269)( 52,265)( 53,267)( 54,266)( 55,298)( 56,300)
( 57,299)( 58,304)( 59,306)( 60,305)( 61,301)( 62,303)( 63,302)( 64,307)
( 65,309)( 66,308)( 67,313)( 68,315)( 69,314)( 70,310)( 71,312)( 72,311)
( 73,316)( 74,318)( 75,317)( 76,322)( 77,324)( 78,323)( 79,319)( 80,321)
( 81,320)( 82,271)( 83,273)( 84,272)( 85,277)( 86,279)( 87,278)( 88,274)
( 89,276)( 90,275)( 91,280)( 92,282)( 93,281)( 94,286)( 95,288)( 96,287)
( 97,283)( 98,285)( 99,284)(100,289)(101,291)(102,290)(103,295)(104,297)
(105,296)(106,292)(107,294)(108,293)(109,379)(110,381)(111,380)(112,385)
(113,387)(114,386)(115,382)(116,384)(117,383)(118,388)(119,390)(120,389)
(121,394)(122,396)(123,395)(124,391)(125,393)(126,392)(127,397)(128,399)
(129,398)(130,403)(131,405)(132,404)(133,400)(134,402)(135,401)(136,406)
(137,408)(138,407)(139,412)(140,414)(141,413)(142,409)(143,411)(144,410)
(145,415)(146,417)(147,416)(148,421)(149,423)(150,422)(151,418)(152,420)
(153,419)(154,424)(155,426)(156,425)(157,430)(158,432)(159,431)(160,427)
(161,429)(162,428)(163,325)(164,327)(165,326)(166,331)(167,333)(168,332)
(169,328)(170,330)(171,329)(172,334)(173,336)(174,335)(175,340)(176,342)
(177,341)(178,337)(179,339)(180,338)(181,343)(182,345)(183,344)(184,349)
(185,351)(186,350)(187,346)(188,348)(189,347)(190,352)(191,354)(192,353)
(193,358)(194,360)(195,359)(196,355)(197,357)(198,356)(199,361)(200,363)
(201,362)(202,367)(203,369)(204,368)(205,364)(206,366)(207,365)(208,370)
(209,372)(210,371)(211,376)(212,378)(213,377)(214,373)(215,375)(216,374);;
s1 := (  1,329)(  2,328)(  3,330)(  4,326)(  5,325)(  6,327)(  7,332)(  8,331)
(  9,333)( 10,347)( 11,346)( 12,348)( 13,344)( 14,343)( 15,345)( 16,350)
( 17,349)( 18,351)( 19,338)( 20,337)( 21,339)( 22,335)( 23,334)( 24,336)
( 25,341)( 26,340)( 27,342)( 28,356)( 29,355)( 30,357)( 31,353)( 32,352)
( 33,354)( 34,359)( 35,358)( 36,360)( 37,374)( 38,373)( 39,375)( 40,371)
( 41,370)( 42,372)( 43,377)( 44,376)( 45,378)( 46,365)( 47,364)( 48,366)
( 49,362)( 50,361)( 51,363)( 52,368)( 53,367)( 54,369)( 55,410)( 56,409)
( 57,411)( 58,407)( 59,406)( 60,408)( 61,413)( 62,412)( 63,414)( 64,428)
( 65,427)( 66,429)( 67,425)( 68,424)( 69,426)( 70,431)( 71,430)( 72,432)
( 73,419)( 74,418)( 75,420)( 76,416)( 77,415)( 78,417)( 79,422)( 80,421)
( 81,423)( 82,383)( 83,382)( 84,384)( 85,380)( 86,379)( 87,381)( 88,386)
( 89,385)( 90,387)( 91,401)( 92,400)( 93,402)( 94,398)( 95,397)( 96,399)
( 97,404)( 98,403)( 99,405)(100,392)(101,391)(102,393)(103,389)(104,388)
(105,390)(106,395)(107,394)(108,396)(109,221)(110,220)(111,222)(112,218)
(113,217)(114,219)(115,224)(116,223)(117,225)(118,239)(119,238)(120,240)
(121,236)(122,235)(123,237)(124,242)(125,241)(126,243)(127,230)(128,229)
(129,231)(130,227)(131,226)(132,228)(133,233)(134,232)(135,234)(136,248)
(137,247)(138,249)(139,245)(140,244)(141,246)(142,251)(143,250)(144,252)
(145,266)(146,265)(147,267)(148,263)(149,262)(150,264)(151,269)(152,268)
(153,270)(154,257)(155,256)(156,258)(157,254)(158,253)(159,255)(160,260)
(161,259)(162,261)(163,302)(164,301)(165,303)(166,299)(167,298)(168,300)
(169,305)(170,304)(171,306)(172,320)(173,319)(174,321)(175,317)(176,316)
(177,318)(178,323)(179,322)(180,324)(181,311)(182,310)(183,312)(184,308)
(185,307)(186,309)(187,314)(188,313)(189,315)(190,275)(191,274)(192,276)
(193,272)(194,271)(195,273)(196,278)(197,277)(198,279)(199,293)(200,292)
(201,294)(202,290)(203,289)(204,291)(205,296)(206,295)(207,297)(208,284)
(209,283)(210,285)(211,281)(212,280)(213,282)(214,287)(215,286)(216,288);;
s2 := (  1,388)(  2,389)(  3,390)(  4,394)(  5,395)(  6,396)(  7,391)(  8,392)
(  9,393)( 10,379)( 11,380)( 12,381)( 13,385)( 14,386)( 15,387)( 16,382)
( 17,383)( 18,384)( 19,397)( 20,398)( 21,399)( 22,403)( 23,404)( 24,405)
( 25,400)( 26,401)( 27,402)( 28,415)( 29,416)( 30,417)( 31,421)( 32,422)
( 33,423)( 34,418)( 35,419)( 36,420)( 37,406)( 38,407)( 39,408)( 40,412)
( 41,413)( 42,414)( 43,409)( 44,410)( 45,411)( 46,424)( 47,425)( 48,426)
( 49,430)( 50,431)( 51,432)( 52,427)( 53,428)( 54,429)( 55,361)( 56,362)
( 57,363)( 58,367)( 59,368)( 60,369)( 61,364)( 62,365)( 63,366)( 64,352)
( 65,353)( 66,354)( 67,358)( 68,359)( 69,360)( 70,355)( 71,356)( 72,357)
( 73,370)( 74,371)( 75,372)( 76,376)( 77,377)( 78,378)( 79,373)( 80,374)
( 81,375)( 82,334)( 83,335)( 84,336)( 85,340)( 86,341)( 87,342)( 88,337)
( 89,338)( 90,339)( 91,325)( 92,326)( 93,327)( 94,331)( 95,332)( 96,333)
( 97,328)( 98,329)( 99,330)(100,343)(101,344)(102,345)(103,349)(104,350)
(105,351)(106,346)(107,347)(108,348)(109,226)(110,227)(111,228)(112,232)
(113,233)(114,234)(115,229)(116,230)(117,231)(118,217)(119,218)(120,219)
(121,223)(122,224)(123,225)(124,220)(125,221)(126,222)(127,235)(128,236)
(129,237)(130,241)(131,242)(132,243)(133,238)(134,239)(135,240)(136,253)
(137,254)(138,255)(139,259)(140,260)(141,261)(142,256)(143,257)(144,258)
(145,244)(146,245)(147,246)(148,250)(149,251)(150,252)(151,247)(152,248)
(153,249)(154,262)(155,263)(156,264)(157,268)(158,269)(159,270)(160,265)
(161,266)(162,267)(163,280)(164,281)(165,282)(166,286)(167,287)(168,288)
(169,283)(170,284)(171,285)(172,271)(173,272)(174,273)(175,277)(176,278)
(177,279)(178,274)(179,275)(180,276)(181,289)(182,290)(183,291)(184,295)
(185,296)(186,297)(187,292)(188,293)(189,294)(190,307)(191,308)(192,309)
(193,313)(194,314)(195,315)(196,310)(197,311)(198,312)(199,298)(200,299)
(201,300)(202,304)(203,305)(204,306)(205,301)(206,302)(207,303)(208,316)
(209,317)(210,318)(211,322)(212,323)(213,324)(214,319)(215,320)(216,321);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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*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(432)!(  1,217)(  2,219)(  3,218)(  4,223)(  5,225)(  6,224)(  7,220)
(  8,222)(  9,221)( 10,226)( 11,228)( 12,227)( 13,232)( 14,234)( 15,233)
( 16,229)( 17,231)( 18,230)( 19,235)( 20,237)( 21,236)( 22,241)( 23,243)
( 24,242)( 25,238)( 26,240)( 27,239)( 28,244)( 29,246)( 30,245)( 31,250)
( 32,252)( 33,251)( 34,247)( 35,249)( 36,248)( 37,253)( 38,255)( 39,254)
( 40,259)( 41,261)( 42,260)( 43,256)( 44,258)( 45,257)( 46,262)( 47,264)
( 48,263)( 49,268)( 50,270)( 51,269)( 52,265)( 53,267)( 54,266)( 55,298)
( 56,300)( 57,299)( 58,304)( 59,306)( 60,305)( 61,301)( 62,303)( 63,302)
( 64,307)( 65,309)( 66,308)( 67,313)( 68,315)( 69,314)( 70,310)( 71,312)
( 72,311)( 73,316)( 74,318)( 75,317)( 76,322)( 77,324)( 78,323)( 79,319)
( 80,321)( 81,320)( 82,271)( 83,273)( 84,272)( 85,277)( 86,279)( 87,278)
( 88,274)( 89,276)( 90,275)( 91,280)( 92,282)( 93,281)( 94,286)( 95,288)
( 96,287)( 97,283)( 98,285)( 99,284)(100,289)(101,291)(102,290)(103,295)
(104,297)(105,296)(106,292)(107,294)(108,293)(109,379)(110,381)(111,380)
(112,385)(113,387)(114,386)(115,382)(116,384)(117,383)(118,388)(119,390)
(120,389)(121,394)(122,396)(123,395)(124,391)(125,393)(126,392)(127,397)
(128,399)(129,398)(130,403)(131,405)(132,404)(133,400)(134,402)(135,401)
(136,406)(137,408)(138,407)(139,412)(140,414)(141,413)(142,409)(143,411)
(144,410)(145,415)(146,417)(147,416)(148,421)(149,423)(150,422)(151,418)
(152,420)(153,419)(154,424)(155,426)(156,425)(157,430)(158,432)(159,431)
(160,427)(161,429)(162,428)(163,325)(164,327)(165,326)(166,331)(167,333)
(168,332)(169,328)(170,330)(171,329)(172,334)(173,336)(174,335)(175,340)
(176,342)(177,341)(178,337)(179,339)(180,338)(181,343)(182,345)(183,344)
(184,349)(185,351)(186,350)(187,346)(188,348)(189,347)(190,352)(191,354)
(192,353)(193,358)(194,360)(195,359)(196,355)(197,357)(198,356)(199,361)
(200,363)(201,362)(202,367)(203,369)(204,368)(205,364)(206,366)(207,365)
(208,370)(209,372)(210,371)(211,376)(212,378)(213,377)(214,373)(215,375)
(216,374);
s1 := Sym(432)!(  1,329)(  2,328)(  3,330)(  4,326)(  5,325)(  6,327)(  7,332)
(  8,331)(  9,333)( 10,347)( 11,346)( 12,348)( 13,344)( 14,343)( 15,345)
( 16,350)( 17,349)( 18,351)( 19,338)( 20,337)( 21,339)( 22,335)( 23,334)
( 24,336)( 25,341)( 26,340)( 27,342)( 28,356)( 29,355)( 30,357)( 31,353)
( 32,352)( 33,354)( 34,359)( 35,358)( 36,360)( 37,374)( 38,373)( 39,375)
( 40,371)( 41,370)( 42,372)( 43,377)( 44,376)( 45,378)( 46,365)( 47,364)
( 48,366)( 49,362)( 50,361)( 51,363)( 52,368)( 53,367)( 54,369)( 55,410)
( 56,409)( 57,411)( 58,407)( 59,406)( 60,408)( 61,413)( 62,412)( 63,414)
( 64,428)( 65,427)( 66,429)( 67,425)( 68,424)( 69,426)( 70,431)( 71,430)
( 72,432)( 73,419)( 74,418)( 75,420)( 76,416)( 77,415)( 78,417)( 79,422)
( 80,421)( 81,423)( 82,383)( 83,382)( 84,384)( 85,380)( 86,379)( 87,381)
( 88,386)( 89,385)( 90,387)( 91,401)( 92,400)( 93,402)( 94,398)( 95,397)
( 96,399)( 97,404)( 98,403)( 99,405)(100,392)(101,391)(102,393)(103,389)
(104,388)(105,390)(106,395)(107,394)(108,396)(109,221)(110,220)(111,222)
(112,218)(113,217)(114,219)(115,224)(116,223)(117,225)(118,239)(119,238)
(120,240)(121,236)(122,235)(123,237)(124,242)(125,241)(126,243)(127,230)
(128,229)(129,231)(130,227)(131,226)(132,228)(133,233)(134,232)(135,234)
(136,248)(137,247)(138,249)(139,245)(140,244)(141,246)(142,251)(143,250)
(144,252)(145,266)(146,265)(147,267)(148,263)(149,262)(150,264)(151,269)
(152,268)(153,270)(154,257)(155,256)(156,258)(157,254)(158,253)(159,255)
(160,260)(161,259)(162,261)(163,302)(164,301)(165,303)(166,299)(167,298)
(168,300)(169,305)(170,304)(171,306)(172,320)(173,319)(174,321)(175,317)
(176,316)(177,318)(178,323)(179,322)(180,324)(181,311)(182,310)(183,312)
(184,308)(185,307)(186,309)(187,314)(188,313)(189,315)(190,275)(191,274)
(192,276)(193,272)(194,271)(195,273)(196,278)(197,277)(198,279)(199,293)
(200,292)(201,294)(202,290)(203,289)(204,291)(205,296)(206,295)(207,297)
(208,284)(209,283)(210,285)(211,281)(212,280)(213,282)(214,287)(215,286)
(216,288);
s2 := Sym(432)!(  1,388)(  2,389)(  3,390)(  4,394)(  5,395)(  6,396)(  7,391)
(  8,392)(  9,393)( 10,379)( 11,380)( 12,381)( 13,385)( 14,386)( 15,387)
( 16,382)( 17,383)( 18,384)( 19,397)( 20,398)( 21,399)( 22,403)( 23,404)
( 24,405)( 25,400)( 26,401)( 27,402)( 28,415)( 29,416)( 30,417)( 31,421)
( 32,422)( 33,423)( 34,418)( 35,419)( 36,420)( 37,406)( 38,407)( 39,408)
( 40,412)( 41,413)( 42,414)( 43,409)( 44,410)( 45,411)( 46,424)( 47,425)
( 48,426)( 49,430)( 50,431)( 51,432)( 52,427)( 53,428)( 54,429)( 55,361)
( 56,362)( 57,363)( 58,367)( 59,368)( 60,369)( 61,364)( 62,365)( 63,366)
( 64,352)( 65,353)( 66,354)( 67,358)( 68,359)( 69,360)( 70,355)( 71,356)
( 72,357)( 73,370)( 74,371)( 75,372)( 76,376)( 77,377)( 78,378)( 79,373)
( 80,374)( 81,375)( 82,334)( 83,335)( 84,336)( 85,340)( 86,341)( 87,342)
( 88,337)( 89,338)( 90,339)( 91,325)( 92,326)( 93,327)( 94,331)( 95,332)
( 96,333)( 97,328)( 98,329)( 99,330)(100,343)(101,344)(102,345)(103,349)
(104,350)(105,351)(106,346)(107,347)(108,348)(109,226)(110,227)(111,228)
(112,232)(113,233)(114,234)(115,229)(116,230)(117,231)(118,217)(119,218)
(120,219)(121,223)(122,224)(123,225)(124,220)(125,221)(126,222)(127,235)
(128,236)(129,237)(130,241)(131,242)(132,243)(133,238)(134,239)(135,240)
(136,253)(137,254)(138,255)(139,259)(140,260)(141,261)(142,256)(143,257)
(144,258)(145,244)(146,245)(147,246)(148,250)(149,251)(150,252)(151,247)
(152,248)(153,249)(154,262)(155,263)(156,264)(157,268)(158,269)(159,270)
(160,265)(161,266)(162,267)(163,280)(164,281)(165,282)(166,286)(167,287)
(168,288)(169,283)(170,284)(171,285)(172,271)(173,272)(174,273)(175,277)
(176,278)(177,279)(178,274)(179,275)(180,276)(181,289)(182,290)(183,291)
(184,295)(185,296)(186,297)(187,292)(188,293)(189,294)(190,307)(191,308)
(192,309)(193,313)(194,314)(195,315)(196,310)(197,311)(198,312)(199,298)
(200,299)(201,300)(202,304)(203,305)(204,306)(205,301)(206,302)(207,303)
(208,316)(209,317)(210,318)(211,322)(212,323)(213,324)(214,319)(215,320)
(216,321);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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*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