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# Polytope of Type {6,24}

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

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

```
References : None.
to this polytope