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# Polytope of Type {24,28}

Atlas Canonical Name : {24,28}*1344a
Also Known As : {24,28|2}. if this polytope has another name.
Group : SmallGroup(1344,2773)
Rank : 3
Schlafli Type : {24,28}
Number of vertices, edges, etc : 24, 336, 28
Order of s0s1s2 : 168
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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 : {24,14}*672, {12,28}*672
3-fold quotients : {8,28}*448a
4-fold quotients : {12,14}*336, {6,28}*336a
6-fold quotients : {4,28}*224, {8,14}*224
7-fold quotients : {24,4}*192a
8-fold quotients : {6,14}*168
12-fold quotients : {2,28}*112, {4,14}*112
14-fold quotients : {12,4}*96a, {24,2}*96
21-fold quotients : {8,4}*64a
24-fold quotients : {2,14}*56
28-fold quotients : {12,2}*48, {6,4}*48a
42-fold quotients : {4,4}*32, {8,2}*32
48-fold quotients : {2,7}*28
56-fold quotients : {6,2}*24
84-fold quotients : {2,4}*16, {4,2}*16
112-fold quotients : {3,2}*12
168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)( 29, 36)
( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)( 51, 58)
( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)( 73, 80)
( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 85,106)( 86,107)( 87,108)( 88,109)
( 89,110)( 90,111)( 91,112)( 92,120)( 93,121)( 94,122)( 95,123)( 96,124)
( 97,125)( 98,126)( 99,113)(100,114)(101,115)(102,116)(103,117)(104,118)
(105,119)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)
(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(141,155)
(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(169,253)(170,254)
(171,255)(172,256)(173,257)(174,258)(175,259)(176,267)(177,268)(178,269)
(179,270)(180,271)(181,272)(182,273)(183,260)(184,261)(185,262)(186,263)
(187,264)(188,265)(189,266)(190,274)(191,275)(192,276)(193,277)(194,278)
(195,279)(196,280)(197,288)(198,289)(199,290)(200,291)(201,292)(202,293)
(203,294)(204,281)(205,282)(206,283)(207,284)(208,285)(209,286)(210,287)
(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)(217,301)(218,309)
(219,310)(220,311)(221,312)(222,313)(223,314)(224,315)(225,302)(226,303)
(227,304)(228,305)(229,306)(230,307)(231,308)(232,316)(233,317)(234,318)
(235,319)(236,320)(237,321)(238,322)(239,330)(240,331)(241,332)(242,333)
(243,334)(244,335)(245,336)(246,323)(247,324)(248,325)(249,326)(250,327)
(251,328)(252,329);;
s1 := (  1,176)(  2,182)(  3,181)(  4,180)(  5,179)(  6,178)(  7,177)(  8,169)
(  9,175)( 10,174)( 11,173)( 12,172)( 13,171)( 14,170)( 15,183)( 16,189)
( 17,188)( 18,187)( 19,186)( 20,185)( 21,184)( 22,197)( 23,203)( 24,202)
( 25,201)( 26,200)( 27,199)( 28,198)( 29,190)( 30,196)( 31,195)( 32,194)
( 33,193)( 34,192)( 35,191)( 36,204)( 37,210)( 38,209)( 39,208)( 40,207)
( 41,206)( 42,205)( 43,218)( 44,224)( 45,223)( 46,222)( 47,221)( 48,220)
( 49,219)( 50,211)( 51,217)( 52,216)( 53,215)( 54,214)( 55,213)( 56,212)
( 57,225)( 58,231)( 59,230)( 60,229)( 61,228)( 62,227)( 63,226)( 64,239)
( 65,245)( 66,244)( 67,243)( 68,242)( 69,241)( 70,240)( 71,232)( 72,238)
( 73,237)( 74,236)( 75,235)( 76,234)( 77,233)( 78,246)( 79,252)( 80,251)
( 81,250)( 82,249)( 83,248)( 84,247)( 85,281)( 86,287)( 87,286)( 88,285)
( 89,284)( 90,283)( 91,282)( 92,274)( 93,280)( 94,279)( 95,278)( 96,277)
( 97,276)( 98,275)( 99,288)(100,294)(101,293)(102,292)(103,291)(104,290)
(105,289)(106,260)(107,266)(108,265)(109,264)(110,263)(111,262)(112,261)
(113,253)(114,259)(115,258)(116,257)(117,256)(118,255)(119,254)(120,267)
(121,273)(122,272)(123,271)(124,270)(125,269)(126,268)(127,323)(128,329)
(129,328)(130,327)(131,326)(132,325)(133,324)(134,316)(135,322)(136,321)
(137,320)(138,319)(139,318)(140,317)(141,330)(142,336)(143,335)(144,334)
(145,333)(146,332)(147,331)(148,302)(149,308)(150,307)(151,306)(152,305)
(153,304)(154,303)(155,295)(156,301)(157,300)(158,299)(159,298)(160,297)
(161,296)(162,309)(163,315)(164,314)(165,313)(166,312)(167,311)(168,310);;
s2 := (  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)
( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)( 36, 37)
( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)( 53, 55)
( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)( 73, 77)
( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)( 92, 93)
( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)(109,111)
(113,114)(115,119)(116,118)(120,121)(122,126)(123,125)(127,128)(129,133)
(130,132)(134,135)(136,140)(137,139)(141,142)(143,147)(144,146)(148,149)
(150,154)(151,153)(155,156)(157,161)(158,160)(162,163)(164,168)(165,167)
(169,212)(170,211)(171,217)(172,216)(173,215)(174,214)(175,213)(176,219)
(177,218)(178,224)(179,223)(180,222)(181,221)(182,220)(183,226)(184,225)
(185,231)(186,230)(187,229)(188,228)(189,227)(190,233)(191,232)(192,238)
(193,237)(194,236)(195,235)(196,234)(197,240)(198,239)(199,245)(200,244)
(201,243)(202,242)(203,241)(204,247)(205,246)(206,252)(207,251)(208,250)
(209,249)(210,248)(253,296)(254,295)(255,301)(256,300)(257,299)(258,298)
(259,297)(260,303)(261,302)(262,308)(263,307)(264,306)(265,305)(266,304)
(267,310)(268,309)(269,315)(270,314)(271,313)(272,312)(273,311)(274,317)
(275,316)(276,322)(277,321)(278,320)(279,319)(280,318)(281,324)(282,323)
(283,329)(284,328)(285,327)(286,326)(287,325)(288,331)(289,330)(290,336)
(291,335)(292,334)(293,333)(294,332);;
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*s2*s1*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,
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*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(336)!(  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)
( 29, 36)( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)
( 51, 58)( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 85,106)( 86,107)( 87,108)
( 88,109)( 89,110)( 90,111)( 91,112)( 92,120)( 93,121)( 94,122)( 95,123)
( 96,124)( 97,125)( 98,126)( 99,113)(100,114)(101,115)(102,116)(103,117)
(104,118)(105,119)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)
(133,154)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)
(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(169,253)
(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,267)(177,268)
(178,269)(179,270)(180,271)(181,272)(182,273)(183,260)(184,261)(185,262)
(186,263)(187,264)(188,265)(189,266)(190,274)(191,275)(192,276)(193,277)
(194,278)(195,279)(196,280)(197,288)(198,289)(199,290)(200,291)(201,292)
(202,293)(203,294)(204,281)(205,282)(206,283)(207,284)(208,285)(209,286)
(210,287)(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)(217,301)
(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)(224,315)(225,302)
(226,303)(227,304)(228,305)(229,306)(230,307)(231,308)(232,316)(233,317)
(234,318)(235,319)(236,320)(237,321)(238,322)(239,330)(240,331)(241,332)
(242,333)(243,334)(244,335)(245,336)(246,323)(247,324)(248,325)(249,326)
(250,327)(251,328)(252,329);
s1 := Sym(336)!(  1,176)(  2,182)(  3,181)(  4,180)(  5,179)(  6,178)(  7,177)
(  8,169)(  9,175)( 10,174)( 11,173)( 12,172)( 13,171)( 14,170)( 15,183)
( 16,189)( 17,188)( 18,187)( 19,186)( 20,185)( 21,184)( 22,197)( 23,203)
( 24,202)( 25,201)( 26,200)( 27,199)( 28,198)( 29,190)( 30,196)( 31,195)
( 32,194)( 33,193)( 34,192)( 35,191)( 36,204)( 37,210)( 38,209)( 39,208)
( 40,207)( 41,206)( 42,205)( 43,218)( 44,224)( 45,223)( 46,222)( 47,221)
( 48,220)( 49,219)( 50,211)( 51,217)( 52,216)( 53,215)( 54,214)( 55,213)
( 56,212)( 57,225)( 58,231)( 59,230)( 60,229)( 61,228)( 62,227)( 63,226)
( 64,239)( 65,245)( 66,244)( 67,243)( 68,242)( 69,241)( 70,240)( 71,232)
( 72,238)( 73,237)( 74,236)( 75,235)( 76,234)( 77,233)( 78,246)( 79,252)
( 80,251)( 81,250)( 82,249)( 83,248)( 84,247)( 85,281)( 86,287)( 87,286)
( 88,285)( 89,284)( 90,283)( 91,282)( 92,274)( 93,280)( 94,279)( 95,278)
( 96,277)( 97,276)( 98,275)( 99,288)(100,294)(101,293)(102,292)(103,291)
(104,290)(105,289)(106,260)(107,266)(108,265)(109,264)(110,263)(111,262)
(112,261)(113,253)(114,259)(115,258)(116,257)(117,256)(118,255)(119,254)
(120,267)(121,273)(122,272)(123,271)(124,270)(125,269)(126,268)(127,323)
(128,329)(129,328)(130,327)(131,326)(132,325)(133,324)(134,316)(135,322)
(136,321)(137,320)(138,319)(139,318)(140,317)(141,330)(142,336)(143,335)
(144,334)(145,333)(146,332)(147,331)(148,302)(149,308)(150,307)(151,306)
(152,305)(153,304)(154,303)(155,295)(156,301)(157,300)(158,299)(159,298)
(160,297)(161,296)(162,309)(163,315)(164,314)(165,313)(166,312)(167,311)
(168,310);
s2 := Sym(336)!(  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)
( 17, 21)( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)
( 36, 37)( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)
( 53, 55)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)
( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)
( 92, 93)( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)
(109,111)(113,114)(115,119)(116,118)(120,121)(122,126)(123,125)(127,128)
(129,133)(130,132)(134,135)(136,140)(137,139)(141,142)(143,147)(144,146)
(148,149)(150,154)(151,153)(155,156)(157,161)(158,160)(162,163)(164,168)
(165,167)(169,212)(170,211)(171,217)(172,216)(173,215)(174,214)(175,213)
(176,219)(177,218)(178,224)(179,223)(180,222)(181,221)(182,220)(183,226)
(184,225)(185,231)(186,230)(187,229)(188,228)(189,227)(190,233)(191,232)
(192,238)(193,237)(194,236)(195,235)(196,234)(197,240)(198,239)(199,245)
(200,244)(201,243)(202,242)(203,241)(204,247)(205,246)(206,252)(207,251)
(208,250)(209,249)(210,248)(253,296)(254,295)(255,301)(256,300)(257,299)
(258,298)(259,297)(260,303)(261,302)(262,308)(263,307)(264,306)(265,305)
(266,304)(267,310)(268,309)(269,315)(270,314)(271,313)(272,312)(273,311)
(274,317)(275,316)(276,322)(277,321)(278,320)(279,319)(280,318)(281,324)
(282,323)(283,329)(284,328)(285,327)(286,326)(287,325)(288,331)(289,330)
(290,336)(291,335)(292,334)(293,333)(294,332);
poly := sub<Sym(336)|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*s2*s1*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,
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*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope