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# Polytope of Type {2,6,24}

Atlas Canonical Name : {2,6,24}*1728f
if this polytope has a name.
Group : SmallGroup(1728,36607)
Rank : 4
Schlafli Type : {2,6,24}
Number of vertices, edges, etc : 2, 18, 216, 72
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 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 : {2,6,12}*864g
3-fold quotients : {2,6,24}*576a, {2,6,24}*576b, {2,6,24}*576c
4-fold quotients : {2,6,6}*432d
6-fold quotients : {2,6,12}*288a, {2,6,12}*288b, {2,6,12}*288c
9-fold quotients : {2,2,24}*192, {2,6,8}*192
12-fold quotients : {2,6,6}*144a, {2,6,6}*144b, {2,6,6}*144c
18-fold quotients : {2,2,12}*96, {2,6,4}*96a
24-fold quotients : {2,3,6}*72, {2,6,3}*72
27-fold quotients : {2,2,8}*64
36-fold quotients : {2,2,6}*48, {2,6,2}*48
54-fold quotients : {2,2,4}*32
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  3,381)(  4,382)(  5,383)(  6,387)(  7,388)(  8,389)(  9,384)( 10,385)
( 11,386)( 12,399)( 13,400)( 14,401)( 15,405)( 16,406)( 17,407)( 18,402)
( 19,403)( 20,404)( 21,390)( 22,391)( 23,392)( 24,396)( 25,397)( 26,398)
( 27,393)( 28,394)( 29,395)( 30,408)( 31,409)( 32,410)( 33,414)( 34,415)
( 35,416)( 36,411)( 37,412)( 38,413)( 39,426)( 40,427)( 41,428)( 42,432)
( 43,433)( 44,434)( 45,429)( 46,430)( 47,431)( 48,417)( 49,418)( 50,419)
( 51,423)( 52,424)( 53,425)( 54,420)( 55,421)( 56,422)( 57,354)( 58,355)
( 59,356)( 60,360)( 61,361)( 62,362)( 63,357)( 64,358)( 65,359)( 66,372)
( 67,373)( 68,374)( 69,378)( 70,379)( 71,380)( 72,375)( 73,376)( 74,377)
( 75,363)( 76,364)( 77,365)( 78,369)( 79,370)( 80,371)( 81,366)( 82,367)
( 83,368)( 84,327)( 85,328)( 86,329)( 87,333)( 88,334)( 89,335)( 90,330)
( 91,331)( 92,332)( 93,345)( 94,346)( 95,347)( 96,351)( 97,352)( 98,353)
( 99,348)(100,349)(101,350)(102,336)(103,337)(104,338)(105,342)(106,343)
(107,344)(108,339)(109,340)(110,341)(111,219)(112,220)(113,221)(114,225)
(115,226)(116,227)(117,222)(118,223)(119,224)(120,237)(121,238)(122,239)
(123,243)(124,244)(125,245)(126,240)(127,241)(128,242)(129,228)(130,229)
(131,230)(132,234)(133,235)(134,236)(135,231)(136,232)(137,233)(138,246)
(139,247)(140,248)(141,252)(142,253)(143,254)(144,249)(145,250)(146,251)
(147,264)(148,265)(149,266)(150,270)(151,271)(152,272)(153,267)(154,268)
(155,269)(156,255)(157,256)(158,257)(159,261)(160,262)(161,263)(162,258)
(163,259)(164,260)(165,273)(166,274)(167,275)(168,279)(169,280)(170,281)
(171,276)(172,277)(173,278)(174,291)(175,292)(176,293)(177,297)(178,298)
(179,299)(180,294)(181,295)(182,296)(183,282)(184,283)(185,284)(186,288)
(187,289)(188,290)(189,285)(190,286)(191,287)(192,300)(193,301)(194,302)
(195,306)(196,307)(197,308)(198,303)(199,304)(200,305)(201,318)(202,319)
(203,320)(204,324)(205,325)(206,326)(207,321)(208,322)(209,323)(210,309)
(211,310)(212,311)(213,315)(214,316)(215,317)(216,312)(217,313)(218,314);;
s2 := (  3,339)(  4,341)(  5,340)(  6,336)(  7,338)(  8,337)(  9,342)( 10,344)
( 11,343)( 12,330)( 13,332)( 14,331)( 15,327)( 16,329)( 17,328)( 18,333)
( 19,335)( 20,334)( 21,348)( 22,350)( 23,349)( 24,345)( 25,347)( 26,346)
( 27,351)( 28,353)( 29,352)( 30,366)( 31,368)( 32,367)( 33,363)( 34,365)
( 35,364)( 36,369)( 37,371)( 38,370)( 39,357)( 40,359)( 41,358)( 42,354)
( 43,356)( 44,355)( 45,360)( 46,362)( 47,361)( 48,375)( 49,377)( 50,376)
( 51,372)( 52,374)( 53,373)( 54,378)( 55,380)( 56,379)( 57,420)( 58,422)
( 59,421)( 60,417)( 61,419)( 62,418)( 63,423)( 64,425)( 65,424)( 66,411)
( 67,413)( 68,412)( 69,408)( 70,410)( 71,409)( 72,414)( 73,416)( 74,415)
( 75,429)( 76,431)( 77,430)( 78,426)( 79,428)( 80,427)( 81,432)( 82,434)
( 83,433)( 84,393)( 85,395)( 86,394)( 87,390)( 88,392)( 89,391)( 90,396)
( 91,398)( 92,397)( 93,384)( 94,386)( 95,385)( 96,381)( 97,383)( 98,382)
( 99,387)(100,389)(101,388)(102,402)(103,404)(104,403)(105,399)(106,401)
(107,400)(108,405)(109,407)(110,406)(111,231)(112,233)(113,232)(114,228)
(115,230)(116,229)(117,234)(118,236)(119,235)(120,222)(121,224)(122,223)
(123,219)(124,221)(125,220)(126,225)(127,227)(128,226)(129,240)(130,242)
(131,241)(132,237)(133,239)(134,238)(135,243)(136,245)(137,244)(138,258)
(139,260)(140,259)(141,255)(142,257)(143,256)(144,261)(145,263)(146,262)
(147,249)(148,251)(149,250)(150,246)(151,248)(152,247)(153,252)(154,254)
(155,253)(156,267)(157,269)(158,268)(159,264)(160,266)(161,265)(162,270)
(163,272)(164,271)(165,312)(166,314)(167,313)(168,309)(169,311)(170,310)
(171,315)(172,317)(173,316)(174,303)(175,305)(176,304)(177,300)(178,302)
(179,301)(180,306)(181,308)(182,307)(183,321)(184,323)(185,322)(186,318)
(187,320)(188,319)(189,324)(190,326)(191,325)(192,285)(193,287)(194,286)
(195,282)(196,284)(197,283)(198,288)(199,290)(200,289)(201,276)(202,278)
(203,277)(204,273)(205,275)(206,274)(207,279)(208,281)(209,280)(210,294)
(211,296)(212,295)(213,291)(214,293)(215,292)(216,297)(217,299)(218,298);;
s3 := (  3,220)(  4,219)(  5,221)(  6,226)(  7,225)(  8,227)(  9,223)( 10,222)
( 11,224)( 12,229)( 13,228)( 14,230)( 15,235)( 16,234)( 17,236)( 18,232)
( 19,231)( 20,233)( 21,238)( 22,237)( 23,239)( 24,244)( 25,243)( 26,245)
( 27,241)( 28,240)( 29,242)( 30,247)( 31,246)( 32,248)( 33,253)( 34,252)
( 35,254)( 36,250)( 37,249)( 38,251)( 39,256)( 40,255)( 41,257)( 42,262)
( 43,261)( 44,263)( 45,259)( 46,258)( 47,260)( 48,265)( 49,264)( 50,266)
( 51,271)( 52,270)( 53,272)( 54,268)( 55,267)( 56,269)( 57,301)( 58,300)
( 59,302)( 60,307)( 61,306)( 62,308)( 63,304)( 64,303)( 65,305)( 66,310)
( 67,309)( 68,311)( 69,316)( 70,315)( 71,317)( 72,313)( 73,312)( 74,314)
( 75,319)( 76,318)( 77,320)( 78,325)( 79,324)( 80,326)( 81,322)( 82,321)
( 83,323)( 84,274)( 85,273)( 86,275)( 87,280)( 88,279)( 89,281)( 90,277)
( 91,276)( 92,278)( 93,283)( 94,282)( 95,284)( 96,289)( 97,288)( 98,290)
( 99,286)(100,285)(101,287)(102,292)(103,291)(104,293)(105,298)(106,297)
(107,299)(108,295)(109,294)(110,296)(111,382)(112,381)(113,383)(114,388)
(115,387)(116,389)(117,385)(118,384)(119,386)(120,391)(121,390)(122,392)
(123,397)(124,396)(125,398)(126,394)(127,393)(128,395)(129,400)(130,399)
(131,401)(132,406)(133,405)(134,407)(135,403)(136,402)(137,404)(138,409)
(139,408)(140,410)(141,415)(142,414)(143,416)(144,412)(145,411)(146,413)
(147,418)(148,417)(149,419)(150,424)(151,423)(152,425)(153,421)(154,420)
(155,422)(156,427)(157,426)(158,428)(159,433)(160,432)(161,434)(162,430)
(163,429)(164,431)(165,328)(166,327)(167,329)(168,334)(169,333)(170,335)
(171,331)(172,330)(173,332)(174,337)(175,336)(176,338)(177,343)(178,342)
(179,344)(180,340)(181,339)(182,341)(183,346)(184,345)(185,347)(186,352)
(187,351)(188,353)(189,349)(190,348)(191,350)(192,355)(193,354)(194,356)
(195,361)(196,360)(197,362)(198,358)(199,357)(200,359)(201,364)(202,363)
(203,365)(204,370)(205,369)(206,371)(207,367)(208,366)(209,368)(210,373)
(211,372)(212,374)(213,379)(214,378)(215,380)(216,376)(217,375)(218,377);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(434)!(1,2);
s1 := Sym(434)!(  3,381)(  4,382)(  5,383)(  6,387)(  7,388)(  8,389)(  9,384)
( 10,385)( 11,386)( 12,399)( 13,400)( 14,401)( 15,405)( 16,406)( 17,407)
( 18,402)( 19,403)( 20,404)( 21,390)( 22,391)( 23,392)( 24,396)( 25,397)
( 26,398)( 27,393)( 28,394)( 29,395)( 30,408)( 31,409)( 32,410)( 33,414)
( 34,415)( 35,416)( 36,411)( 37,412)( 38,413)( 39,426)( 40,427)( 41,428)
( 42,432)( 43,433)( 44,434)( 45,429)( 46,430)( 47,431)( 48,417)( 49,418)
( 50,419)( 51,423)( 52,424)( 53,425)( 54,420)( 55,421)( 56,422)( 57,354)
( 58,355)( 59,356)( 60,360)( 61,361)( 62,362)( 63,357)( 64,358)( 65,359)
( 66,372)( 67,373)( 68,374)( 69,378)( 70,379)( 71,380)( 72,375)( 73,376)
( 74,377)( 75,363)( 76,364)( 77,365)( 78,369)( 79,370)( 80,371)( 81,366)
( 82,367)( 83,368)( 84,327)( 85,328)( 86,329)( 87,333)( 88,334)( 89,335)
( 90,330)( 91,331)( 92,332)( 93,345)( 94,346)( 95,347)( 96,351)( 97,352)
( 98,353)( 99,348)(100,349)(101,350)(102,336)(103,337)(104,338)(105,342)
(106,343)(107,344)(108,339)(109,340)(110,341)(111,219)(112,220)(113,221)
(114,225)(115,226)(116,227)(117,222)(118,223)(119,224)(120,237)(121,238)
(122,239)(123,243)(124,244)(125,245)(126,240)(127,241)(128,242)(129,228)
(130,229)(131,230)(132,234)(133,235)(134,236)(135,231)(136,232)(137,233)
(138,246)(139,247)(140,248)(141,252)(142,253)(143,254)(144,249)(145,250)
(146,251)(147,264)(148,265)(149,266)(150,270)(151,271)(152,272)(153,267)
(154,268)(155,269)(156,255)(157,256)(158,257)(159,261)(160,262)(161,263)
(162,258)(163,259)(164,260)(165,273)(166,274)(167,275)(168,279)(169,280)
(170,281)(171,276)(172,277)(173,278)(174,291)(175,292)(176,293)(177,297)
(178,298)(179,299)(180,294)(181,295)(182,296)(183,282)(184,283)(185,284)
(186,288)(187,289)(188,290)(189,285)(190,286)(191,287)(192,300)(193,301)
(194,302)(195,306)(196,307)(197,308)(198,303)(199,304)(200,305)(201,318)
(202,319)(203,320)(204,324)(205,325)(206,326)(207,321)(208,322)(209,323)
(210,309)(211,310)(212,311)(213,315)(214,316)(215,317)(216,312)(217,313)
(218,314);
s2 := Sym(434)!(  3,339)(  4,341)(  5,340)(  6,336)(  7,338)(  8,337)(  9,342)
( 10,344)( 11,343)( 12,330)( 13,332)( 14,331)( 15,327)( 16,329)( 17,328)
( 18,333)( 19,335)( 20,334)( 21,348)( 22,350)( 23,349)( 24,345)( 25,347)
( 26,346)( 27,351)( 28,353)( 29,352)( 30,366)( 31,368)( 32,367)( 33,363)
( 34,365)( 35,364)( 36,369)( 37,371)( 38,370)( 39,357)( 40,359)( 41,358)
( 42,354)( 43,356)( 44,355)( 45,360)( 46,362)( 47,361)( 48,375)( 49,377)
( 50,376)( 51,372)( 52,374)( 53,373)( 54,378)( 55,380)( 56,379)( 57,420)
( 58,422)( 59,421)( 60,417)( 61,419)( 62,418)( 63,423)( 64,425)( 65,424)
( 66,411)( 67,413)( 68,412)( 69,408)( 70,410)( 71,409)( 72,414)( 73,416)
( 74,415)( 75,429)( 76,431)( 77,430)( 78,426)( 79,428)( 80,427)( 81,432)
( 82,434)( 83,433)( 84,393)( 85,395)( 86,394)( 87,390)( 88,392)( 89,391)
( 90,396)( 91,398)( 92,397)( 93,384)( 94,386)( 95,385)( 96,381)( 97,383)
( 98,382)( 99,387)(100,389)(101,388)(102,402)(103,404)(104,403)(105,399)
(106,401)(107,400)(108,405)(109,407)(110,406)(111,231)(112,233)(113,232)
(114,228)(115,230)(116,229)(117,234)(118,236)(119,235)(120,222)(121,224)
(122,223)(123,219)(124,221)(125,220)(126,225)(127,227)(128,226)(129,240)
(130,242)(131,241)(132,237)(133,239)(134,238)(135,243)(136,245)(137,244)
(138,258)(139,260)(140,259)(141,255)(142,257)(143,256)(144,261)(145,263)
(146,262)(147,249)(148,251)(149,250)(150,246)(151,248)(152,247)(153,252)
(154,254)(155,253)(156,267)(157,269)(158,268)(159,264)(160,266)(161,265)
(162,270)(163,272)(164,271)(165,312)(166,314)(167,313)(168,309)(169,311)
(170,310)(171,315)(172,317)(173,316)(174,303)(175,305)(176,304)(177,300)
(178,302)(179,301)(180,306)(181,308)(182,307)(183,321)(184,323)(185,322)
(186,318)(187,320)(188,319)(189,324)(190,326)(191,325)(192,285)(193,287)
(194,286)(195,282)(196,284)(197,283)(198,288)(199,290)(200,289)(201,276)
(202,278)(203,277)(204,273)(205,275)(206,274)(207,279)(208,281)(209,280)
(210,294)(211,296)(212,295)(213,291)(214,293)(215,292)(216,297)(217,299)
(218,298);
s3 := Sym(434)!(  3,220)(  4,219)(  5,221)(  6,226)(  7,225)(  8,227)(  9,223)
( 10,222)( 11,224)( 12,229)( 13,228)( 14,230)( 15,235)( 16,234)( 17,236)
( 18,232)( 19,231)( 20,233)( 21,238)( 22,237)( 23,239)( 24,244)( 25,243)
( 26,245)( 27,241)( 28,240)( 29,242)( 30,247)( 31,246)( 32,248)( 33,253)
( 34,252)( 35,254)( 36,250)( 37,249)( 38,251)( 39,256)( 40,255)( 41,257)
( 42,262)( 43,261)( 44,263)( 45,259)( 46,258)( 47,260)( 48,265)( 49,264)
( 50,266)( 51,271)( 52,270)( 53,272)( 54,268)( 55,267)( 56,269)( 57,301)
( 58,300)( 59,302)( 60,307)( 61,306)( 62,308)( 63,304)( 64,303)( 65,305)
( 66,310)( 67,309)( 68,311)( 69,316)( 70,315)( 71,317)( 72,313)( 73,312)
( 74,314)( 75,319)( 76,318)( 77,320)( 78,325)( 79,324)( 80,326)( 81,322)
( 82,321)( 83,323)( 84,274)( 85,273)( 86,275)( 87,280)( 88,279)( 89,281)
( 90,277)( 91,276)( 92,278)( 93,283)( 94,282)( 95,284)( 96,289)( 97,288)
( 98,290)( 99,286)(100,285)(101,287)(102,292)(103,291)(104,293)(105,298)
(106,297)(107,299)(108,295)(109,294)(110,296)(111,382)(112,381)(113,383)
(114,388)(115,387)(116,389)(117,385)(118,384)(119,386)(120,391)(121,390)
(122,392)(123,397)(124,396)(125,398)(126,394)(127,393)(128,395)(129,400)
(130,399)(131,401)(132,406)(133,405)(134,407)(135,403)(136,402)(137,404)
(138,409)(139,408)(140,410)(141,415)(142,414)(143,416)(144,412)(145,411)
(146,413)(147,418)(148,417)(149,419)(150,424)(151,423)(152,425)(153,421)
(154,420)(155,422)(156,427)(157,426)(158,428)(159,433)(160,432)(161,434)
(162,430)(163,429)(164,431)(165,328)(166,327)(167,329)(168,334)(169,333)
(170,335)(171,331)(172,330)(173,332)(174,337)(175,336)(176,338)(177,343)
(178,342)(179,344)(180,340)(181,339)(182,341)(183,346)(184,345)(185,347)
(186,352)(187,351)(188,353)(189,349)(190,348)(191,350)(192,355)(193,354)
(194,356)(195,361)(196,360)(197,362)(198,358)(199,357)(200,359)(201,364)
(202,363)(203,365)(204,370)(205,369)(206,371)(207,367)(208,366)(209,368)
(210,373)(211,372)(212,374)(213,379)(214,378)(215,380)(216,376)(217,375)
(218,377);
poly := sub<Sym(434)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope