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# Polytope of Type {12,56}

Atlas Canonical Name : {12,56}*1344b
if this polytope has a name.
Group : SmallGroup(1344,2918)
Rank : 3
Schlafli Type : {12,56}
Number of vertices, edges, etc : 12, 336, 56
Order of s0s1s2 : 168
Order of s0s1s2s1 : 4
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 : {12,28}*672
3-fold quotients : {4,56}*448b
4-fold quotients : {12,14}*336, {6,28}*336a
6-fold quotients : {4,28}*224
7-fold quotients : {12,8}*192b
8-fold quotients : {6,14}*168
12-fold quotients : {2,28}*112, {4,14}*112
14-fold quotients : {12,4}*96a
21-fold quotients : {4,8}*64b
24-fold quotients : {2,14}*56
28-fold quotients : {12,2}*48, {6,4}*48a
42-fold quotients : {4,4}*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)( 43, 64)( 44, 65)
( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 78)( 51, 79)( 52, 80)
( 53, 81)( 54, 82)( 55, 83)( 56, 84)( 57, 71)( 58, 72)( 59, 73)( 60, 74)
( 61, 75)( 62, 76)( 63, 77)( 92, 99)( 93,100)( 94,101)( 95,102)( 96,103)
( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)
(119,126)(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)
(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)(218,330)
(219,331)(220,332)(221,333)(222,334)(223,335)(224,336)(225,323)(226,324)
(227,325)(228,326)(229,327)(230,328)(231,329)(232,295)(233,296)(234,297)
(235,298)(236,299)(237,300)(238,301)(239,309)(240,310)(241,311)(242,312)
(243,313)(244,314)(245,315)(246,302)(247,303)(248,304)(249,305)(250,306)
(251,307)(252,308);;
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)( 43,239)( 44,245)( 45,244)( 46,243)( 47,242)( 48,241)
( 49,240)( 50,232)( 51,238)( 52,237)( 53,236)( 54,235)( 55,234)( 56,233)
( 57,246)( 58,252)( 59,251)( 60,250)( 61,249)( 62,248)( 63,247)( 64,218)
( 65,224)( 66,223)( 67,222)( 68,221)( 69,220)( 70,219)( 71,211)( 72,217)
( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,225)( 79,231)( 80,230)
( 81,229)( 82,228)( 83,227)( 84,226)( 85,260)( 86,266)( 87,265)( 88,264)
( 89,263)( 90,262)( 91,261)( 92,253)( 93,259)( 94,258)( 95,257)( 96,256)
( 97,255)( 98,254)( 99,267)(100,273)(101,272)(102,271)(103,270)(104,269)
(105,268)(106,281)(107,287)(108,286)(109,285)(110,284)(111,283)(112,282)
(113,274)(114,280)(115,279)(116,278)(117,277)(118,276)(119,275)(120,288)
(121,294)(122,293)(123,292)(124,291)(125,290)(126,289)(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, 65)( 44, 64)( 45, 70)( 46, 69)( 47, 68)( 48, 67)
( 49, 66)( 50, 72)( 51, 71)( 52, 77)( 53, 76)( 54, 75)( 55, 74)( 56, 73)
( 57, 79)( 58, 78)( 59, 84)( 60, 83)( 61, 82)( 62, 81)( 63, 80)( 85,107)
( 86,106)( 87,112)( 88,111)( 89,110)( 90,109)( 91,108)( 92,114)( 93,113)
( 94,119)( 95,118)( 96,117)( 97,116)( 98,115)( 99,121)(100,120)(101,126)
(102,125)(103,124)(104,123)(105,122)(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)
(253,317)(254,316)(255,322)(256,321)(257,320)(258,319)(259,318)(260,324)
(261,323)(262,329)(263,328)(264,327)(265,326)(266,325)(267,331)(268,330)
(269,336)(270,335)(271,334)(272,333)(273,332)(274,296)(275,295)(276,301)
(277,300)(278,299)(279,298)(280,297)(281,303)(282,302)(283,308)(284,307)
(285,306)(286,305)(287,304)(288,310)(289,309)(290,315)(291,314)(292,313)
(293,312)(294,311);;
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,
s0*s1*s2*s1*s0*s1*s0*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,
s2*s0*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*s0*s1*s2*s1 ];;
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)( 43, 64)
( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 78)( 51, 79)
( 52, 80)( 53, 81)( 54, 82)( 55, 83)( 56, 84)( 57, 71)( 58, 72)( 59, 73)
( 60, 74)( 61, 75)( 62, 76)( 63, 77)( 92, 99)( 93,100)( 94,101)( 95,102)
( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)(117,124)
(118,125)(119,126)(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)(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)
(218,330)(219,331)(220,332)(221,333)(222,334)(223,335)(224,336)(225,323)
(226,324)(227,325)(228,326)(229,327)(230,328)(231,329)(232,295)(233,296)
(234,297)(235,298)(236,299)(237,300)(238,301)(239,309)(240,310)(241,311)
(242,312)(243,313)(244,314)(245,315)(246,302)(247,303)(248,304)(249,305)
(250,306)(251,307)(252,308);
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)( 43,239)( 44,245)( 45,244)( 46,243)( 47,242)
( 48,241)( 49,240)( 50,232)( 51,238)( 52,237)( 53,236)( 54,235)( 55,234)
( 56,233)( 57,246)( 58,252)( 59,251)( 60,250)( 61,249)( 62,248)( 63,247)
( 64,218)( 65,224)( 66,223)( 67,222)( 68,221)( 69,220)( 70,219)( 71,211)
( 72,217)( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,225)( 79,231)
( 80,230)( 81,229)( 82,228)( 83,227)( 84,226)( 85,260)( 86,266)( 87,265)
( 88,264)( 89,263)( 90,262)( 91,261)( 92,253)( 93,259)( 94,258)( 95,257)
( 96,256)( 97,255)( 98,254)( 99,267)(100,273)(101,272)(102,271)(103,270)
(104,269)(105,268)(106,281)(107,287)(108,286)(109,285)(110,284)(111,283)
(112,282)(113,274)(114,280)(115,279)(116,278)(117,277)(118,276)(119,275)
(120,288)(121,294)(122,293)(123,292)(124,291)(125,290)(126,289)(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, 65)( 44, 64)( 45, 70)( 46, 69)( 47, 68)
( 48, 67)( 49, 66)( 50, 72)( 51, 71)( 52, 77)( 53, 76)( 54, 75)( 55, 74)
( 56, 73)( 57, 79)( 58, 78)( 59, 84)( 60, 83)( 61, 82)( 62, 81)( 63, 80)
( 85,107)( 86,106)( 87,112)( 88,111)( 89,110)( 90,109)( 91,108)( 92,114)
( 93,113)( 94,119)( 95,118)( 96,117)( 97,116)( 98,115)( 99,121)(100,120)
(101,126)(102,125)(103,124)(104,123)(105,122)(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)(253,317)(254,316)(255,322)(256,321)(257,320)(258,319)(259,318)
(260,324)(261,323)(262,329)(263,328)(264,327)(265,326)(266,325)(267,331)
(268,330)(269,336)(270,335)(271,334)(272,333)(273,332)(274,296)(275,295)
(276,301)(277,300)(278,299)(279,298)(280,297)(281,303)(282,302)(283,308)
(284,307)(285,306)(286,305)(287,304)(288,310)(289,309)(290,315)(291,314)
(292,313)(293,312)(294,311);
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*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,
s2*s0*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*s0*s1*s2*s1 >;

```
References : None.
to this polytope