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# Polytope of Type {14,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,42}*1176b
Also Known As : {14,42|2}. if this polytope has another name.
Group : SmallGroup(1176,265)
Rank : 3
Schlafli Type : {14,42}
Number of vertices, edges, etc : 14, 294, 42
Order of s0s1s2 : 42
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) :
3-fold quotients : {14,14}*392a
7-fold quotients : {14,6}*168, {2,42}*168
14-fold quotients : {2,21}*84
21-fold quotients : {2,14}*56, {14,2}*56
42-fold quotients : {2,7}*28, {7,2}*28
49-fold quotients : {2,6}*24
98-fold quotients : {2,3}*12
147-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)
( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)( 73, 76)
( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)
( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)
(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)(129,132)
(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)(149,154)
(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)(165,166)
(170,175)(171,174)(172,173)(177,182)(178,181)(179,180)(184,189)(185,188)
(186,187)(191,196)(192,195)(193,194)(198,203)(199,202)(200,201)(205,210)
(206,209)(207,208)(212,217)(213,216)(214,215)(219,224)(220,223)(221,222)
(226,231)(227,230)(228,229)(233,238)(234,237)(235,236)(240,245)(241,244)
(242,243)(247,252)(248,251)(249,250)(254,259)(255,258)(256,257)(261,266)
(262,265)(263,264)(268,273)(269,272)(270,271)(275,280)(276,279)(277,278)
(282,287)(283,286)(284,285)(289,294)(290,293)(291,292);;
s1 := (  1,  2)(  3,  7)(  4,  6)(  8, 44)(  9, 43)( 10, 49)( 11, 48)( 12, 47)
( 13, 46)( 14, 45)( 15, 37)( 16, 36)( 17, 42)( 18, 41)( 19, 40)( 20, 39)
( 21, 38)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)( 27, 32)( 28, 31)
( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101)( 57,142)
( 58,141)( 59,147)( 60,146)( 61,145)( 62,144)( 63,143)( 64,135)( 65,134)
( 66,140)( 67,139)( 68,138)( 69,137)( 70,136)( 71,128)( 72,127)( 73,133)
( 74,132)( 75,131)( 76,130)( 77,129)( 78,121)( 79,120)( 80,126)( 81,125)
( 82,124)( 83,123)( 84,122)( 85,114)( 86,113)( 87,119)( 88,118)( 89,117)
( 90,116)( 91,115)( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)
( 98,108)(148,149)(150,154)(151,153)(155,191)(156,190)(157,196)(158,195)
(159,194)(160,193)(161,192)(162,184)(163,183)(164,189)(165,188)(166,187)
(167,186)(168,185)(169,177)(170,176)(171,182)(172,181)(173,180)(174,179)
(175,178)(197,247)(198,246)(199,252)(200,251)(201,250)(202,249)(203,248)
(204,289)(205,288)(206,294)(207,293)(208,292)(209,291)(210,290)(211,282)
(212,281)(213,287)(214,286)(215,285)(216,284)(217,283)(218,275)(219,274)
(220,280)(221,279)(222,278)(223,277)(224,276)(225,268)(226,267)(227,273)
(228,272)(229,271)(230,270)(231,269)(232,261)(233,260)(234,266)(235,265)
(236,264)(237,263)(238,262)(239,254)(240,253)(241,259)(242,258)(243,257)
(244,256)(245,255);;
s2 := (  1,204)(  2,205)(  3,206)(  4,207)(  5,208)(  6,209)(  7,210)(  8,197)
(  9,198)( 10,199)( 11,200)( 12,201)( 13,202)( 14,203)( 15,239)( 16,240)
( 17,241)( 18,242)( 19,243)( 20,244)( 21,245)( 22,232)( 23,233)( 24,234)
( 25,235)( 26,236)( 27,237)( 28,238)( 29,225)( 30,226)( 31,227)( 32,228)
( 33,229)( 34,230)( 35,231)( 36,218)( 37,219)( 38,220)( 39,221)( 40,222)
( 41,223)( 42,224)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)( 48,216)
( 49,217)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)( 56,161)
( 57,148)( 58,149)( 59,150)( 60,151)( 61,152)( 62,153)( 63,154)( 64,190)
( 65,191)( 66,192)( 67,193)( 68,194)( 69,195)( 70,196)( 71,183)( 72,184)
( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,176)( 79,177)( 80,178)
( 81,179)( 82,180)( 83,181)( 84,182)( 85,169)( 86,170)( 87,171)( 88,172)
( 89,173)( 90,174)( 91,175)( 92,162)( 93,163)( 94,164)( 95,165)( 96,166)
( 97,167)( 98,168)( 99,253)(100,254)(101,255)(102,256)(103,257)(104,258)
(105,259)(106,246)(107,247)(108,248)(109,249)(110,250)(111,251)(112,252)
(113,288)(114,289)(115,290)(116,291)(117,292)(118,293)(119,294)(120,281)
(121,282)(122,283)(123,284)(124,285)(125,286)(126,287)(127,274)(128,275)
(129,276)(130,277)(131,278)(132,279)(133,280)(134,267)(135,268)(136,269)
(137,270)(138,271)(139,272)(140,273)(141,260)(142,261)(143,262)(144,263)
(145,264)(146,265)(147,266);;
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,
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*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(294)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)
( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)
( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)
( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)
( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)
(109,110)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)
(129,132)(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)
(149,154)(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)
(165,166)(170,175)(171,174)(172,173)(177,182)(178,181)(179,180)(184,189)
(185,188)(186,187)(191,196)(192,195)(193,194)(198,203)(199,202)(200,201)
(205,210)(206,209)(207,208)(212,217)(213,216)(214,215)(219,224)(220,223)
(221,222)(226,231)(227,230)(228,229)(233,238)(234,237)(235,236)(240,245)
(241,244)(242,243)(247,252)(248,251)(249,250)(254,259)(255,258)(256,257)
(261,266)(262,265)(263,264)(268,273)(269,272)(270,271)(275,280)(276,279)
(277,278)(282,287)(283,286)(284,285)(289,294)(290,293)(291,292);
s1 := Sym(294)!(  1,  2)(  3,  7)(  4,  6)(  8, 44)(  9, 43)( 10, 49)( 11, 48)
( 12, 47)( 13, 46)( 14, 45)( 15, 37)( 16, 36)( 17, 42)( 18, 41)( 19, 40)
( 20, 39)( 21, 38)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)( 27, 32)
( 28, 31)( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101)
( 57,142)( 58,141)( 59,147)( 60,146)( 61,145)( 62,144)( 63,143)( 64,135)
( 65,134)( 66,140)( 67,139)( 68,138)( 69,137)( 70,136)( 71,128)( 72,127)
( 73,133)( 74,132)( 75,131)( 76,130)( 77,129)( 78,121)( 79,120)( 80,126)
( 81,125)( 82,124)( 83,123)( 84,122)( 85,114)( 86,113)( 87,119)( 88,118)
( 89,117)( 90,116)( 91,115)( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)
( 97,109)( 98,108)(148,149)(150,154)(151,153)(155,191)(156,190)(157,196)
(158,195)(159,194)(160,193)(161,192)(162,184)(163,183)(164,189)(165,188)
(166,187)(167,186)(168,185)(169,177)(170,176)(171,182)(172,181)(173,180)
(174,179)(175,178)(197,247)(198,246)(199,252)(200,251)(201,250)(202,249)
(203,248)(204,289)(205,288)(206,294)(207,293)(208,292)(209,291)(210,290)
(211,282)(212,281)(213,287)(214,286)(215,285)(216,284)(217,283)(218,275)
(219,274)(220,280)(221,279)(222,278)(223,277)(224,276)(225,268)(226,267)
(227,273)(228,272)(229,271)(230,270)(231,269)(232,261)(233,260)(234,266)
(235,265)(236,264)(237,263)(238,262)(239,254)(240,253)(241,259)(242,258)
(243,257)(244,256)(245,255);
s2 := Sym(294)!(  1,204)(  2,205)(  3,206)(  4,207)(  5,208)(  6,209)(  7,210)
(  8,197)(  9,198)( 10,199)( 11,200)( 12,201)( 13,202)( 14,203)( 15,239)
( 16,240)( 17,241)( 18,242)( 19,243)( 20,244)( 21,245)( 22,232)( 23,233)
( 24,234)( 25,235)( 26,236)( 27,237)( 28,238)( 29,225)( 30,226)( 31,227)
( 32,228)( 33,229)( 34,230)( 35,231)( 36,218)( 37,219)( 38,220)( 39,221)
( 40,222)( 41,223)( 42,224)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)
( 48,216)( 49,217)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)
( 56,161)( 57,148)( 58,149)( 59,150)( 60,151)( 61,152)( 62,153)( 63,154)
( 64,190)( 65,191)( 66,192)( 67,193)( 68,194)( 69,195)( 70,196)( 71,183)
( 72,184)( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,176)( 79,177)
( 80,178)( 81,179)( 82,180)( 83,181)( 84,182)( 85,169)( 86,170)( 87,171)
( 88,172)( 89,173)( 90,174)( 91,175)( 92,162)( 93,163)( 94,164)( 95,165)
( 96,166)( 97,167)( 98,168)( 99,253)(100,254)(101,255)(102,256)(103,257)
(104,258)(105,259)(106,246)(107,247)(108,248)(109,249)(110,250)(111,251)
(112,252)(113,288)(114,289)(115,290)(116,291)(117,292)(118,293)(119,294)
(120,281)(121,282)(122,283)(123,284)(124,285)(125,286)(126,287)(127,274)
(128,275)(129,276)(130,277)(131,278)(132,279)(133,280)(134,267)(135,268)
(136,269)(137,270)(138,271)(139,272)(140,273)(141,260)(142,261)(143,262)
(144,263)(145,264)(146,265)(147,266);
poly := sub<Sym(294)|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,
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*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