# Polytope of Type {4,102}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,102}*816c
if this polytope has a name.
Group : SmallGroup(816,195)
Rank : 3
Schlafli Type : {4,102}
Number of vertices, edges, etc : 4, 204, 102
Order of s0s1s2 : 51
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,102,2} of size 1632
Vertex Figure Of :
{2,4,102} of size 1632
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,51}*408
17-fold quotients : {4,6}*48b
34-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,102}*1632
Permutation Representation (GAP) :
```s0 := (  1,207)(  2,208)(  3,205)(  4,206)(  5,211)(  6,212)(  7,209)(  8,210)
(  9,215)( 10,216)( 11,213)( 12,214)( 13,219)( 14,220)( 15,217)( 16,218)
( 17,223)( 18,224)( 19,221)( 20,222)( 21,227)( 22,228)( 23,225)( 24,226)
( 25,231)( 26,232)( 27,229)( 28,230)( 29,235)( 30,236)( 31,233)( 32,234)
( 33,239)( 34,240)( 35,237)( 36,238)( 37,243)( 38,244)( 39,241)( 40,242)
( 41,247)( 42,248)( 43,245)( 44,246)( 45,251)( 46,252)( 47,249)( 48,250)
( 49,255)( 50,256)( 51,253)( 52,254)( 53,259)( 54,260)( 55,257)( 56,258)
( 57,263)( 58,264)( 59,261)( 60,262)( 61,267)( 62,268)( 63,265)( 64,266)
( 65,271)( 66,272)( 67,269)( 68,270)( 69,275)( 70,276)( 71,273)( 72,274)
( 73,279)( 74,280)( 75,277)( 76,278)( 77,283)( 78,284)( 79,281)( 80,282)
( 81,287)( 82,288)( 83,285)( 84,286)( 85,291)( 86,292)( 87,289)( 88,290)
( 89,295)( 90,296)( 91,293)( 92,294)( 93,299)( 94,300)( 95,297)( 96,298)
( 97,303)( 98,304)( 99,301)(100,302)(101,307)(102,308)(103,305)(104,306)
(105,311)(106,312)(107,309)(108,310)(109,315)(110,316)(111,313)(112,314)
(113,319)(114,320)(115,317)(116,318)(117,323)(118,324)(119,321)(120,322)
(121,327)(122,328)(123,325)(124,326)(125,331)(126,332)(127,329)(128,330)
(129,335)(130,336)(131,333)(132,334)(133,339)(134,340)(135,337)(136,338)
(137,343)(138,344)(139,341)(140,342)(141,347)(142,348)(143,345)(144,346)
(145,351)(146,352)(147,349)(148,350)(149,355)(150,356)(151,353)(152,354)
(153,359)(154,360)(155,357)(156,358)(157,363)(158,364)(159,361)(160,362)
(161,367)(162,368)(163,365)(164,366)(165,371)(166,372)(167,369)(168,370)
(169,375)(170,376)(171,373)(172,374)(173,379)(174,380)(175,377)(176,378)
(177,383)(178,384)(179,381)(180,382)(181,387)(182,388)(183,385)(184,386)
(185,391)(186,392)(187,389)(188,390)(189,395)(190,396)(191,393)(192,394)
(193,399)(194,400)(195,397)(196,398)(197,403)(198,404)(199,401)(200,402)
(201,407)(202,408)(203,405)(204,406);;
s1 := (  2,  3)(  5, 65)(  6, 67)(  7, 66)(  8, 68)(  9, 61)( 10, 63)( 11, 62)
( 12, 64)( 13, 57)( 14, 59)( 15, 58)( 16, 60)( 17, 53)( 18, 55)( 19, 54)
( 20, 56)( 21, 49)( 22, 51)( 23, 50)( 24, 52)( 25, 45)( 26, 47)( 27, 46)
( 28, 48)( 29, 41)( 30, 43)( 31, 42)( 32, 44)( 33, 37)( 34, 39)( 35, 38)
( 36, 40)( 69,137)( 70,139)( 71,138)( 72,140)( 73,201)( 74,203)( 75,202)
( 76,204)( 77,197)( 78,199)( 79,198)( 80,200)( 81,193)( 82,195)( 83,194)
( 84,196)( 85,189)( 86,191)( 87,190)( 88,192)( 89,185)( 90,187)( 91,186)
( 92,188)( 93,181)( 94,183)( 95,182)( 96,184)( 97,177)( 98,179)( 99,178)
(100,180)(101,173)(102,175)(103,174)(104,176)(105,169)(106,171)(107,170)
(108,172)(109,165)(110,167)(111,166)(112,168)(113,161)(114,163)(115,162)
(116,164)(117,157)(118,159)(119,158)(120,160)(121,153)(122,155)(123,154)
(124,156)(125,149)(126,151)(127,150)(128,152)(129,145)(130,147)(131,146)
(132,148)(133,141)(134,143)(135,142)(136,144)(206,207)(209,269)(210,271)
(211,270)(212,272)(213,265)(214,267)(215,266)(216,268)(217,261)(218,263)
(219,262)(220,264)(221,257)(222,259)(223,258)(224,260)(225,253)(226,255)
(227,254)(228,256)(229,249)(230,251)(231,250)(232,252)(233,245)(234,247)
(235,246)(236,248)(237,241)(238,243)(239,242)(240,244)(273,341)(274,343)
(275,342)(276,344)(277,405)(278,407)(279,406)(280,408)(281,401)(282,403)
(283,402)(284,404)(285,397)(286,399)(287,398)(288,400)(289,393)(290,395)
(291,394)(292,396)(293,389)(294,391)(295,390)(296,392)(297,385)(298,387)
(299,386)(300,388)(301,381)(302,383)(303,382)(304,384)(305,377)(306,379)
(307,378)(308,380)(309,373)(310,375)(311,374)(312,376)(313,369)(314,371)
(315,370)(316,372)(317,365)(318,367)(319,366)(320,368)(321,361)(322,363)
(323,362)(324,364)(325,357)(326,359)(327,358)(328,360)(329,353)(330,355)
(331,354)(332,356)(333,349)(334,351)(335,350)(336,352)(337,345)(338,347)
(339,346)(340,348);;
s2 := (  1,345)(  2,348)(  3,347)(  4,346)(  5,341)(  6,344)(  7,343)(  8,342)
(  9,405)( 10,408)( 11,407)( 12,406)( 13,401)( 14,404)( 15,403)( 16,402)
( 17,397)( 18,400)( 19,399)( 20,398)( 21,393)( 22,396)( 23,395)( 24,394)
( 25,389)( 26,392)( 27,391)( 28,390)( 29,385)( 30,388)( 31,387)( 32,386)
( 33,381)( 34,384)( 35,383)( 36,382)( 37,377)( 38,380)( 39,379)( 40,378)
( 41,373)( 42,376)( 43,375)( 44,374)( 45,369)( 46,372)( 47,371)( 48,370)
( 49,365)( 50,368)( 51,367)( 52,366)( 53,361)( 54,364)( 55,363)( 56,362)
( 57,357)( 58,360)( 59,359)( 60,358)( 61,353)( 62,356)( 63,355)( 64,354)
( 65,349)( 66,352)( 67,351)( 68,350)( 69,277)( 70,280)( 71,279)( 72,278)
( 73,273)( 74,276)( 75,275)( 76,274)( 77,337)( 78,340)( 79,339)( 80,338)
( 81,333)( 82,336)( 83,335)( 84,334)( 85,329)( 86,332)( 87,331)( 88,330)
( 89,325)( 90,328)( 91,327)( 92,326)( 93,321)( 94,324)( 95,323)( 96,322)
( 97,317)( 98,320)( 99,319)(100,318)(101,313)(102,316)(103,315)(104,314)
(105,309)(106,312)(107,311)(108,310)(109,305)(110,308)(111,307)(112,306)
(113,301)(114,304)(115,303)(116,302)(117,297)(118,300)(119,299)(120,298)
(121,293)(122,296)(123,295)(124,294)(125,289)(126,292)(127,291)(128,290)
(129,285)(130,288)(131,287)(132,286)(133,281)(134,284)(135,283)(136,282)
(137,209)(138,212)(139,211)(140,210)(141,205)(142,208)(143,207)(144,206)
(145,269)(146,272)(147,271)(148,270)(149,265)(150,268)(151,267)(152,266)
(153,261)(154,264)(155,263)(156,262)(157,257)(158,260)(159,259)(160,258)
(161,253)(162,256)(163,255)(164,254)(165,249)(166,252)(167,251)(168,250)
(169,245)(170,248)(171,247)(172,246)(173,241)(174,244)(175,243)(176,242)
(177,237)(178,240)(179,239)(180,238)(181,233)(182,236)(183,235)(184,234)
(185,229)(186,232)(187,231)(188,230)(189,225)(190,228)(191,227)(192,226)
(193,221)(194,224)(195,223)(196,222)(197,217)(198,220)(199,219)(200,218)
(201,213)(202,216)(203,215)(204,214);;
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*s2*s1*s0*s1*s2*s1*s2*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*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(408)!(  1,207)(  2,208)(  3,205)(  4,206)(  5,211)(  6,212)(  7,209)
(  8,210)(  9,215)( 10,216)( 11,213)( 12,214)( 13,219)( 14,220)( 15,217)
( 16,218)( 17,223)( 18,224)( 19,221)( 20,222)( 21,227)( 22,228)( 23,225)
( 24,226)( 25,231)( 26,232)( 27,229)( 28,230)( 29,235)( 30,236)( 31,233)
( 32,234)( 33,239)( 34,240)( 35,237)( 36,238)( 37,243)( 38,244)( 39,241)
( 40,242)( 41,247)( 42,248)( 43,245)( 44,246)( 45,251)( 46,252)( 47,249)
( 48,250)( 49,255)( 50,256)( 51,253)( 52,254)( 53,259)( 54,260)( 55,257)
( 56,258)( 57,263)( 58,264)( 59,261)( 60,262)( 61,267)( 62,268)( 63,265)
( 64,266)( 65,271)( 66,272)( 67,269)( 68,270)( 69,275)( 70,276)( 71,273)
( 72,274)( 73,279)( 74,280)( 75,277)( 76,278)( 77,283)( 78,284)( 79,281)
( 80,282)( 81,287)( 82,288)( 83,285)( 84,286)( 85,291)( 86,292)( 87,289)
( 88,290)( 89,295)( 90,296)( 91,293)( 92,294)( 93,299)( 94,300)( 95,297)
( 96,298)( 97,303)( 98,304)( 99,301)(100,302)(101,307)(102,308)(103,305)
(104,306)(105,311)(106,312)(107,309)(108,310)(109,315)(110,316)(111,313)
(112,314)(113,319)(114,320)(115,317)(116,318)(117,323)(118,324)(119,321)
(120,322)(121,327)(122,328)(123,325)(124,326)(125,331)(126,332)(127,329)
(128,330)(129,335)(130,336)(131,333)(132,334)(133,339)(134,340)(135,337)
(136,338)(137,343)(138,344)(139,341)(140,342)(141,347)(142,348)(143,345)
(144,346)(145,351)(146,352)(147,349)(148,350)(149,355)(150,356)(151,353)
(152,354)(153,359)(154,360)(155,357)(156,358)(157,363)(158,364)(159,361)
(160,362)(161,367)(162,368)(163,365)(164,366)(165,371)(166,372)(167,369)
(168,370)(169,375)(170,376)(171,373)(172,374)(173,379)(174,380)(175,377)
(176,378)(177,383)(178,384)(179,381)(180,382)(181,387)(182,388)(183,385)
(184,386)(185,391)(186,392)(187,389)(188,390)(189,395)(190,396)(191,393)
(192,394)(193,399)(194,400)(195,397)(196,398)(197,403)(198,404)(199,401)
(200,402)(201,407)(202,408)(203,405)(204,406);
s1 := Sym(408)!(  2,  3)(  5, 65)(  6, 67)(  7, 66)(  8, 68)(  9, 61)( 10, 63)
( 11, 62)( 12, 64)( 13, 57)( 14, 59)( 15, 58)( 16, 60)( 17, 53)( 18, 55)
( 19, 54)( 20, 56)( 21, 49)( 22, 51)( 23, 50)( 24, 52)( 25, 45)( 26, 47)
( 27, 46)( 28, 48)( 29, 41)( 30, 43)( 31, 42)( 32, 44)( 33, 37)( 34, 39)
( 35, 38)( 36, 40)( 69,137)( 70,139)( 71,138)( 72,140)( 73,201)( 74,203)
( 75,202)( 76,204)( 77,197)( 78,199)( 79,198)( 80,200)( 81,193)( 82,195)
( 83,194)( 84,196)( 85,189)( 86,191)( 87,190)( 88,192)( 89,185)( 90,187)
( 91,186)( 92,188)( 93,181)( 94,183)( 95,182)( 96,184)( 97,177)( 98,179)
( 99,178)(100,180)(101,173)(102,175)(103,174)(104,176)(105,169)(106,171)
(107,170)(108,172)(109,165)(110,167)(111,166)(112,168)(113,161)(114,163)
(115,162)(116,164)(117,157)(118,159)(119,158)(120,160)(121,153)(122,155)
(123,154)(124,156)(125,149)(126,151)(127,150)(128,152)(129,145)(130,147)
(131,146)(132,148)(133,141)(134,143)(135,142)(136,144)(206,207)(209,269)
(210,271)(211,270)(212,272)(213,265)(214,267)(215,266)(216,268)(217,261)
(218,263)(219,262)(220,264)(221,257)(222,259)(223,258)(224,260)(225,253)
(226,255)(227,254)(228,256)(229,249)(230,251)(231,250)(232,252)(233,245)
(234,247)(235,246)(236,248)(237,241)(238,243)(239,242)(240,244)(273,341)
(274,343)(275,342)(276,344)(277,405)(278,407)(279,406)(280,408)(281,401)
(282,403)(283,402)(284,404)(285,397)(286,399)(287,398)(288,400)(289,393)
(290,395)(291,394)(292,396)(293,389)(294,391)(295,390)(296,392)(297,385)
(298,387)(299,386)(300,388)(301,381)(302,383)(303,382)(304,384)(305,377)
(306,379)(307,378)(308,380)(309,373)(310,375)(311,374)(312,376)(313,369)
(314,371)(315,370)(316,372)(317,365)(318,367)(319,366)(320,368)(321,361)
(322,363)(323,362)(324,364)(325,357)(326,359)(327,358)(328,360)(329,353)
(330,355)(331,354)(332,356)(333,349)(334,351)(335,350)(336,352)(337,345)
(338,347)(339,346)(340,348);
s2 := Sym(408)!(  1,345)(  2,348)(  3,347)(  4,346)(  5,341)(  6,344)(  7,343)
(  8,342)(  9,405)( 10,408)( 11,407)( 12,406)( 13,401)( 14,404)( 15,403)
( 16,402)( 17,397)( 18,400)( 19,399)( 20,398)( 21,393)( 22,396)( 23,395)
( 24,394)( 25,389)( 26,392)( 27,391)( 28,390)( 29,385)( 30,388)( 31,387)
( 32,386)( 33,381)( 34,384)( 35,383)( 36,382)( 37,377)( 38,380)( 39,379)
( 40,378)( 41,373)( 42,376)( 43,375)( 44,374)( 45,369)( 46,372)( 47,371)
( 48,370)( 49,365)( 50,368)( 51,367)( 52,366)( 53,361)( 54,364)( 55,363)
( 56,362)( 57,357)( 58,360)( 59,359)( 60,358)( 61,353)( 62,356)( 63,355)
( 64,354)( 65,349)( 66,352)( 67,351)( 68,350)( 69,277)( 70,280)( 71,279)
( 72,278)( 73,273)( 74,276)( 75,275)( 76,274)( 77,337)( 78,340)( 79,339)
( 80,338)( 81,333)( 82,336)( 83,335)( 84,334)( 85,329)( 86,332)( 87,331)
( 88,330)( 89,325)( 90,328)( 91,327)( 92,326)( 93,321)( 94,324)( 95,323)
( 96,322)( 97,317)( 98,320)( 99,319)(100,318)(101,313)(102,316)(103,315)
(104,314)(105,309)(106,312)(107,311)(108,310)(109,305)(110,308)(111,307)
(112,306)(113,301)(114,304)(115,303)(116,302)(117,297)(118,300)(119,299)
(120,298)(121,293)(122,296)(123,295)(124,294)(125,289)(126,292)(127,291)
(128,290)(129,285)(130,288)(131,287)(132,286)(133,281)(134,284)(135,283)
(136,282)(137,209)(138,212)(139,211)(140,210)(141,205)(142,208)(143,207)
(144,206)(145,269)(146,272)(147,271)(148,270)(149,265)(150,268)(151,267)
(152,266)(153,261)(154,264)(155,263)(156,262)(157,257)(158,260)(159,259)
(160,258)(161,253)(162,256)(163,255)(164,254)(165,249)(166,252)(167,251)
(168,250)(169,245)(170,248)(171,247)(172,246)(173,241)(174,244)(175,243)
(176,242)(177,237)(178,240)(179,239)(180,238)(181,233)(182,236)(183,235)
(184,234)(185,229)(186,232)(187,231)(188,230)(189,225)(190,228)(191,227)
(192,226)(193,221)(194,224)(195,223)(196,222)(197,217)(198,220)(199,219)
(200,218)(201,213)(202,216)(203,215)(204,214);
poly := sub<Sym(408)|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*s2*s1*s0*s1*s2*s1*s2*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*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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