# Polytope of Type {4,81}

Atlas Canonical Name : {4,81}*648
if this polytope has a name.
Group : SmallGroup(648,41)
Rank : 3
Schlafli Type : {4,81}
Number of vertices, edges, etc : 4, 162, 81
Order of s0s1s2 : 81
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,81,2} of size 1296
Vertex Figure Of :
{2,4,81} of size 1296
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,27}*216
9-fold quotients : {4,9}*72
27-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,81}*1296, {4,162}*1296b, {4,162}*1296c
3-fold covers : {4,243}*1944
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);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 37, 97)( 38, 99)( 39, 98)( 40,100)( 41,105)( 42,107)( 43,106)
( 44,108)( 45,101)( 46,103)( 47,102)( 48,104)( 49, 85)( 50, 87)( 51, 86)
( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)
( 60, 92)( 61, 73)( 62, 75)( 63, 74)( 64, 76)( 65, 81)( 66, 83)( 67, 82)
( 68, 84)( 69, 77)( 70, 79)( 71, 78)( 72, 80)(109,313)(110,315)(111,314)
(112,316)(113,321)(114,323)(115,322)(116,324)(117,317)(118,319)(119,318)
(120,320)(121,301)(122,303)(123,302)(124,304)(125,309)(126,311)(127,310)
(128,312)(129,305)(130,307)(131,306)(132,308)(133,289)(134,291)(135,290)
(136,292)(137,297)(138,299)(139,298)(140,300)(141,293)(142,295)(143,294)
(144,296)(145,277)(146,279)(147,278)(148,280)(149,285)(150,287)(151,286)
(152,288)(153,281)(154,283)(155,282)(156,284)(157,265)(158,267)(159,266)
(160,268)(161,273)(162,275)(163,274)(164,276)(165,269)(166,271)(167,270)
(168,272)(169,253)(170,255)(171,254)(172,256)(173,261)(174,263)(175,262)
(176,264)(177,257)(178,259)(179,258)(180,260)(181,241)(182,243)(183,242)
(184,244)(185,249)(186,251)(187,250)(188,252)(189,245)(190,247)(191,246)
(192,248)(193,229)(194,231)(195,230)(196,232)(197,237)(198,239)(199,238)
(200,240)(201,233)(202,235)(203,234)(204,236)(205,217)(206,219)(207,218)
(208,220)(209,225)(210,227)(211,226)(212,228)(213,221)(214,223)(215,222)
(216,224);;
s2 := (  1,217)(  2,220)(  3,219)(  4,218)(  5,225)(  6,228)(  7,227)(  8,226)
(  9,221)( 10,224)( 11,223)( 12,222)( 13,245)( 14,248)( 15,247)( 16,246)
( 17,241)( 18,244)( 19,243)( 20,242)( 21,249)( 22,252)( 23,251)( 24,250)
( 25,233)( 26,236)( 27,235)( 28,234)( 29,229)( 30,232)( 31,231)( 32,230)
( 33,237)( 34,240)( 35,239)( 36,238)( 37,313)( 38,316)( 39,315)( 40,314)
( 41,321)( 42,324)( 43,323)( 44,322)( 45,317)( 46,320)( 47,319)( 48,318)
( 49,301)( 50,304)( 51,303)( 52,302)( 53,309)( 54,312)( 55,311)( 56,310)
( 57,305)( 58,308)( 59,307)( 60,306)( 61,289)( 62,292)( 63,291)( 64,290)
( 65,297)( 66,300)( 67,299)( 68,298)( 69,293)( 70,296)( 71,295)( 72,294)
( 73,277)( 74,280)( 75,279)( 76,278)( 77,285)( 78,288)( 79,287)( 80,286)
( 81,281)( 82,284)( 83,283)( 84,282)( 85,265)( 86,268)( 87,267)( 88,266)
( 89,273)( 90,276)( 91,275)( 92,274)( 93,269)( 94,272)( 95,271)( 96,270)
( 97,253)( 98,256)( 99,255)(100,254)(101,261)(102,264)(103,263)(104,262)
(105,257)(106,260)(107,259)(108,258)(110,112)(113,117)(114,120)(115,119)
(116,118)(121,137)(122,140)(123,139)(124,138)(125,133)(126,136)(127,135)
(128,134)(129,141)(130,144)(131,143)(132,142)(145,205)(146,208)(147,207)
(148,206)(149,213)(150,216)(151,215)(152,214)(153,209)(154,212)(155,211)
(156,210)(157,193)(158,196)(159,195)(160,194)(161,201)(162,204)(163,203)
(164,202)(165,197)(166,200)(167,199)(168,198)(169,181)(170,184)(171,183)
(172,182)(173,189)(174,192)(175,191)(176,190)(177,185)(178,188)(179,187)
(180,186);;
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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope