Questions?
See the FAQ
or other info.

# Polytope of Type {6,84}

Atlas Canonical Name : {6,84}*1344a
if this polytope has a name.
Group : SmallGroup(1344,11328)
Rank : 3
Schlafli Type : {6,84}
Number of vertices, edges, etc : 8, 336, 112
Order of s0s1s2 : 28
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,42}*672
4-fold quotients : {6,21}*336
7-fold quotients : {6,12}*192a
12-fold quotients : {2,28}*112
14-fold quotients : {6,6}*96
24-fold quotients : {2,14}*56
28-fold quotients : {3,6}*48, {6,3}*48
48-fold quotients : {2,7}*28
56-fold quotients : {3,3}*24
84-fold quotients : {2,4}*16
168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 29, 57)
( 30, 58)( 31, 60)( 32, 59)( 33, 61)( 34, 62)( 35, 64)( 36, 63)( 37, 65)
( 38, 66)( 39, 68)( 40, 67)( 41, 69)( 42, 70)( 43, 72)( 44, 71)( 45, 73)
( 46, 74)( 47, 76)( 48, 75)( 49, 77)( 50, 78)( 51, 80)( 52, 79)( 53, 81)
( 54, 82)( 55, 84)( 56, 83)( 87, 88)( 91, 92)( 95, 96)( 99,100)(103,104)
(107,108)(111,112)(113,141)(114,142)(115,144)(116,143)(117,145)(118,146)
(119,148)(120,147)(121,149)(122,150)(123,152)(124,151)(125,153)(126,154)
(127,156)(128,155)(129,157)(130,158)(131,160)(132,159)(133,161)(134,162)
(135,164)(136,163)(137,165)(138,166)(139,168)(140,167)(171,172)(175,176)
(179,180)(183,184)(187,188)(191,192)(195,196)(197,225)(198,226)(199,228)
(200,227)(201,229)(202,230)(203,232)(204,231)(205,233)(206,234)(207,236)
(208,235)(209,237)(210,238)(211,240)(212,239)(213,241)(214,242)(215,244)
(216,243)(217,245)(218,246)(219,248)(220,247)(221,249)(222,250)(223,252)
(224,251)(255,256)(259,260)(263,264)(267,268)(271,272)(275,276)(279,280)
(281,309)(282,310)(283,312)(284,311)(285,313)(286,314)(287,316)(288,315)
(289,317)(290,318)(291,320)(292,319)(293,321)(294,322)(295,324)(296,323)
(297,325)(298,326)(299,328)(300,327)(301,329)(302,330)(303,332)(304,331)
(305,333)(306,334)(307,336)(308,335);;
s1 := (  1, 29)(  2, 32)(  3, 31)(  4, 30)(  5, 53)(  6, 56)(  7, 55)(  8, 54)
(  9, 49)( 10, 52)( 11, 51)( 12, 50)( 13, 45)( 14, 48)( 15, 47)( 16, 46)
( 17, 41)( 18, 44)( 19, 43)( 20, 42)( 21, 37)( 22, 40)( 23, 39)( 24, 38)
( 25, 33)( 26, 36)( 27, 35)( 28, 34)( 58, 60)( 61, 81)( 62, 84)( 63, 83)
( 64, 82)( 65, 77)( 66, 80)( 67, 79)( 68, 78)( 69, 73)( 70, 76)( 71, 75)
( 72, 74)( 85,113)( 86,116)( 87,115)( 88,114)( 89,137)( 90,140)( 91,139)
( 92,138)( 93,133)( 94,136)( 95,135)( 96,134)( 97,129)( 98,132)( 99,131)
(100,130)(101,125)(102,128)(103,127)(104,126)(105,121)(106,124)(107,123)
(108,122)(109,117)(110,120)(111,119)(112,118)(142,144)(145,165)(146,168)
(147,167)(148,166)(149,161)(150,164)(151,163)(152,162)(153,157)(154,160)
(155,159)(156,158)(169,281)(170,284)(171,283)(172,282)(173,305)(174,308)
(175,307)(176,306)(177,301)(178,304)(179,303)(180,302)(181,297)(182,300)
(183,299)(184,298)(185,293)(186,296)(187,295)(188,294)(189,289)(190,292)
(191,291)(192,290)(193,285)(194,288)(195,287)(196,286)(197,253)(198,256)
(199,255)(200,254)(201,277)(202,280)(203,279)(204,278)(205,273)(206,276)
(207,275)(208,274)(209,269)(210,272)(211,271)(212,270)(213,265)(214,268)
(215,267)(216,266)(217,261)(218,264)(219,263)(220,262)(221,257)(222,260)
(223,259)(224,258)(225,309)(226,312)(227,311)(228,310)(229,333)(230,336)
(231,335)(232,334)(233,329)(234,332)(235,331)(236,330)(237,325)(238,328)
(239,327)(240,326)(241,321)(242,324)(243,323)(244,322)(245,317)(246,320)
(247,319)(248,318)(249,313)(250,316)(251,315)(252,314);;
s2 := (  1,174)(  2,173)(  3,175)(  4,176)(  5,170)(  6,169)(  7,171)(  8,172)
(  9,194)( 10,193)( 11,195)( 12,196)( 13,190)( 14,189)( 15,191)( 16,192)
( 17,186)( 18,185)( 19,187)( 20,188)( 21,182)( 22,181)( 23,183)( 24,184)
( 25,178)( 26,177)( 27,179)( 28,180)( 29,230)( 30,229)( 31,231)( 32,232)
( 33,226)( 34,225)( 35,227)( 36,228)( 37,250)( 38,249)( 39,251)( 40,252)
( 41,246)( 42,245)( 43,247)( 44,248)( 45,242)( 46,241)( 47,243)( 48,244)
( 49,238)( 50,237)( 51,239)( 52,240)( 53,234)( 54,233)( 55,235)( 56,236)
( 57,202)( 58,201)( 59,203)( 60,204)( 61,198)( 62,197)( 63,199)( 64,200)
( 65,222)( 66,221)( 67,223)( 68,224)( 69,218)( 70,217)( 71,219)( 72,220)
( 73,214)( 74,213)( 75,215)( 76,216)( 77,210)( 78,209)( 79,211)( 80,212)
( 81,206)( 82,205)( 83,207)( 84,208)( 85,258)( 86,257)( 87,259)( 88,260)
( 89,254)( 90,253)( 91,255)( 92,256)( 93,278)( 94,277)( 95,279)( 96,280)
( 97,274)( 98,273)( 99,275)(100,276)(101,270)(102,269)(103,271)(104,272)
(105,266)(106,265)(107,267)(108,268)(109,262)(110,261)(111,263)(112,264)
(113,314)(114,313)(115,315)(116,316)(117,310)(118,309)(119,311)(120,312)
(121,334)(122,333)(123,335)(124,336)(125,330)(126,329)(127,331)(128,332)
(129,326)(130,325)(131,327)(132,328)(133,322)(134,321)(135,323)(136,324)
(137,318)(138,317)(139,319)(140,320)(141,286)(142,285)(143,287)(144,288)
(145,282)(146,281)(147,283)(148,284)(149,306)(150,305)(151,307)(152,308)
(153,302)(154,301)(155,303)(156,304)(157,298)(158,297)(159,299)(160,300)
(161,294)(162,293)(163,295)(164,296)(165,290)(166,289)(167,291)(168,292);;
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*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*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*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope