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# Polytope of Type {4,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,84}*672b
if this polytope has a name.
Group : SmallGroup(672,1080)
Rank : 3
Schlafli Type : {4,84}
Number of vertices, edges, etc : 4, 168, 84
Order of s0s1s2 : 84
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,84,2} of size 1344
Vertex Figure Of :
{2,4,84} of size 1344
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,42}*336b
4-fold quotients : {4,21}*168
7-fold quotients : {4,12}*96b
14-fold quotients : {4,6}*48c
28-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,168}*1344c, {4,168}*1344d, {4,84}*1344b
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336);;
s1 := (  2,  3)(  5, 25)(  6, 27)(  7, 26)(  8, 28)(  9, 21)( 10, 23)( 11, 22)
( 12, 24)( 13, 17)( 14, 19)( 15, 18)( 16, 20)( 29, 57)( 30, 59)( 31, 58)
( 32, 60)( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 77)( 38, 79)( 39, 78)
( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 69)( 46, 71)( 47, 70)
( 48, 72)( 49, 65)( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)
( 56, 64)( 86, 87)( 89,109)( 90,111)( 91,110)( 92,112)( 93,105)( 94,107)
( 95,106)( 96,108)( 97,101)( 98,103)( 99,102)(100,104)(113,141)(114,143)
(115,142)(116,144)(117,165)(118,167)(119,166)(120,168)(121,161)(122,163)
(123,162)(124,164)(125,157)(126,159)(127,158)(128,160)(129,153)(130,155)
(131,154)(132,156)(133,149)(134,151)(135,150)(136,152)(137,145)(138,147)
(139,146)(140,148)(169,253)(170,255)(171,254)(172,256)(173,277)(174,279)
(175,278)(176,280)(177,273)(178,275)(179,274)(180,276)(181,269)(182,271)
(183,270)(184,272)(185,265)(186,267)(187,266)(188,268)(189,261)(190,263)
(191,262)(192,264)(193,257)(194,259)(195,258)(196,260)(197,309)(198,311)
(199,310)(200,312)(201,333)(202,335)(203,334)(204,336)(205,329)(206,331)
(207,330)(208,332)(209,325)(210,327)(211,326)(212,328)(213,321)(214,323)
(215,322)(216,324)(217,317)(218,319)(219,318)(220,320)(221,313)(222,315)
(223,314)(224,316)(225,281)(226,283)(227,282)(228,284)(229,305)(230,307)
(231,306)(232,308)(233,301)(234,303)(235,302)(236,304)(237,297)(238,299)
(239,298)(240,300)(241,293)(242,295)(243,294)(244,296)(245,289)(246,291)
(247,290)(248,292)(249,285)(250,287)(251,286)(252,288);;
s2 := (  1,229)(  2,232)(  3,231)(  4,230)(  5,225)(  6,228)(  7,227)(  8,226)
(  9,249)( 10,252)( 11,251)( 12,250)( 13,245)( 14,248)( 15,247)( 16,246)
( 17,241)( 18,244)( 19,243)( 20,242)( 21,237)( 22,240)( 23,239)( 24,238)
( 25,233)( 26,236)( 27,235)( 28,234)( 29,201)( 30,204)( 31,203)( 32,202)
( 33,197)( 34,200)( 35,199)( 36,198)( 37,221)( 38,224)( 39,223)( 40,222)
( 41,217)( 42,220)( 43,219)( 44,218)( 45,213)( 46,216)( 47,215)( 48,214)
( 49,209)( 50,212)( 51,211)( 52,210)( 53,205)( 54,208)( 55,207)( 56,206)
( 57,173)( 58,176)( 59,175)( 60,174)( 61,169)( 62,172)( 63,171)( 64,170)
( 65,193)( 66,196)( 67,195)( 68,194)( 69,189)( 70,192)( 71,191)( 72,190)
( 73,185)( 74,188)( 75,187)( 76,186)( 77,181)( 78,184)( 79,183)( 80,182)
( 81,177)( 82,180)( 83,179)( 84,178)( 85,313)( 86,316)( 87,315)( 88,314)
( 89,309)( 90,312)( 91,311)( 92,310)( 93,333)( 94,336)( 95,335)( 96,334)
( 97,329)( 98,332)( 99,331)(100,330)(101,325)(102,328)(103,327)(104,326)
(105,321)(106,324)(107,323)(108,322)(109,317)(110,320)(111,319)(112,318)
(113,285)(114,288)(115,287)(116,286)(117,281)(118,284)(119,283)(120,282)
(121,305)(122,308)(123,307)(124,306)(125,301)(126,304)(127,303)(128,302)
(129,297)(130,300)(131,299)(132,298)(133,293)(134,296)(135,295)(136,294)
(137,289)(138,292)(139,291)(140,290)(141,257)(142,260)(143,259)(144,258)
(145,253)(146,256)(147,255)(148,254)(149,277)(150,280)(151,279)(152,278)
(153,273)(154,276)(155,275)(156,274)(157,269)(158,272)(159,271)(160,270)
(161,265)(162,268)(163,267)(164,266)(165,261)(166,264)(167,263)(168,262);;
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*s2*s1*s0*s1*s2*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*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(336)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336);
s1 := Sym(336)!(  2,  3)(  5, 25)(  6, 27)(  7, 26)(  8, 28)(  9, 21)( 10, 23)
( 11, 22)( 12, 24)( 13, 17)( 14, 19)( 15, 18)( 16, 20)( 29, 57)( 30, 59)
( 31, 58)( 32, 60)( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 77)( 38, 79)
( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 69)( 46, 71)
( 47, 70)( 48, 72)( 49, 65)( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)
( 55, 62)( 56, 64)( 86, 87)( 89,109)( 90,111)( 91,110)( 92,112)( 93,105)
( 94,107)( 95,106)( 96,108)( 97,101)( 98,103)( 99,102)(100,104)(113,141)
(114,143)(115,142)(116,144)(117,165)(118,167)(119,166)(120,168)(121,161)
(122,163)(123,162)(124,164)(125,157)(126,159)(127,158)(128,160)(129,153)
(130,155)(131,154)(132,156)(133,149)(134,151)(135,150)(136,152)(137,145)
(138,147)(139,146)(140,148)(169,253)(170,255)(171,254)(172,256)(173,277)
(174,279)(175,278)(176,280)(177,273)(178,275)(179,274)(180,276)(181,269)
(182,271)(183,270)(184,272)(185,265)(186,267)(187,266)(188,268)(189,261)
(190,263)(191,262)(192,264)(193,257)(194,259)(195,258)(196,260)(197,309)
(198,311)(199,310)(200,312)(201,333)(202,335)(203,334)(204,336)(205,329)
(206,331)(207,330)(208,332)(209,325)(210,327)(211,326)(212,328)(213,321)
(214,323)(215,322)(216,324)(217,317)(218,319)(219,318)(220,320)(221,313)
(222,315)(223,314)(224,316)(225,281)(226,283)(227,282)(228,284)(229,305)
(230,307)(231,306)(232,308)(233,301)(234,303)(235,302)(236,304)(237,297)
(238,299)(239,298)(240,300)(241,293)(242,295)(243,294)(244,296)(245,289)
(246,291)(247,290)(248,292)(249,285)(250,287)(251,286)(252,288);
s2 := Sym(336)!(  1,229)(  2,232)(  3,231)(  4,230)(  5,225)(  6,228)(  7,227)
(  8,226)(  9,249)( 10,252)( 11,251)( 12,250)( 13,245)( 14,248)( 15,247)
( 16,246)( 17,241)( 18,244)( 19,243)( 20,242)( 21,237)( 22,240)( 23,239)
( 24,238)( 25,233)( 26,236)( 27,235)( 28,234)( 29,201)( 30,204)( 31,203)
( 32,202)( 33,197)( 34,200)( 35,199)( 36,198)( 37,221)( 38,224)( 39,223)
( 40,222)( 41,217)( 42,220)( 43,219)( 44,218)( 45,213)( 46,216)( 47,215)
( 48,214)( 49,209)( 50,212)( 51,211)( 52,210)( 53,205)( 54,208)( 55,207)
( 56,206)( 57,173)( 58,176)( 59,175)( 60,174)( 61,169)( 62,172)( 63,171)
( 64,170)( 65,193)( 66,196)( 67,195)( 68,194)( 69,189)( 70,192)( 71,191)
( 72,190)( 73,185)( 74,188)( 75,187)( 76,186)( 77,181)( 78,184)( 79,183)
( 80,182)( 81,177)( 82,180)( 83,179)( 84,178)( 85,313)( 86,316)( 87,315)
( 88,314)( 89,309)( 90,312)( 91,311)( 92,310)( 93,333)( 94,336)( 95,335)
( 96,334)( 97,329)( 98,332)( 99,331)(100,330)(101,325)(102,328)(103,327)
(104,326)(105,321)(106,324)(107,323)(108,322)(109,317)(110,320)(111,319)
(112,318)(113,285)(114,288)(115,287)(116,286)(117,281)(118,284)(119,283)
(120,282)(121,305)(122,308)(123,307)(124,306)(125,301)(126,304)(127,303)
(128,302)(129,297)(130,300)(131,299)(132,298)(133,293)(134,296)(135,295)
(136,294)(137,289)(138,292)(139,291)(140,290)(141,257)(142,260)(143,259)
(144,258)(145,253)(146,256)(147,255)(148,254)(149,277)(150,280)(151,279)
(152,278)(153,273)(154,276)(155,275)(156,274)(157,269)(158,272)(159,271)
(160,270)(161,265)(162,268)(163,267)(164,266)(165,261)(166,264)(167,263)
(168,262);
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*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*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