Polytope of Type {2,324}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,324}*1296
if this polytope has a name.
Group : SmallGroup(1296,65)
Rank : 3
Schlafli Type : {2,324}
Number of vertices, edges, etc : 2, 324, 324
Order of s0s1s2 : 324
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
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) :
2-fold quotients : {2,162}*648
3-fold quotients : {2,108}*432
4-fold quotients : {2,81}*324
6-fold quotients : {2,54}*216
9-fold quotients : {2,36}*144
12-fold quotients : {2,27}*108
18-fold quotients : {2,18}*72
27-fold quotients : {2,12}*48
36-fold quotients : {2,9}*36
54-fold quotients : {2,6}*24
81-fold quotients : {2,4}*16
108-fold quotients : {2,3}*12
162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 11)( 7, 10)( 8, 9)( 12, 29)( 13, 28)( 14, 27)( 15, 26)
( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 30, 83)( 31, 82)( 32, 81)
( 33, 80)( 34, 79)( 35, 78)( 36, 77)( 37, 76)( 38, 75)( 39, 74)( 40, 73)
( 41, 72)( 42, 71)( 43, 70)( 44, 69)( 45, 68)( 46, 67)( 47, 66)( 48, 65)
( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 60)( 54, 59)( 55, 58)( 56, 57)
( 85, 86)( 87, 92)( 88, 91)( 89, 90)( 93,110)( 94,109)( 95,108)( 96,107)
( 97,106)( 98,105)( 99,104)(100,103)(101,102)(111,164)(112,163)(113,162)
(114,161)(115,160)(116,159)(117,158)(118,157)(119,156)(120,155)(121,154)
(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)
(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)
(165,246)(166,248)(167,247)(168,254)(169,253)(170,252)(171,251)(172,250)
(173,249)(174,272)(175,271)(176,270)(177,269)(178,268)(179,267)(180,266)
(181,265)(182,264)(183,263)(184,262)(185,261)(186,260)(187,259)(188,258)
(189,257)(190,256)(191,255)(192,326)(193,325)(194,324)(195,323)(196,322)
(197,321)(198,320)(199,319)(200,318)(201,317)(202,316)(203,315)(204,314)
(205,313)(206,312)(207,311)(208,310)(209,309)(210,308)(211,307)(212,306)
(213,305)(214,304)(215,303)(216,302)(217,301)(218,300)(219,299)(220,298)
(221,297)(222,296)(223,295)(224,294)(225,293)(226,292)(227,291)(228,290)
(229,289)(230,288)(231,287)(232,286)(233,285)(234,284)(235,283)(236,282)
(237,281)(238,280)(239,279)(240,278)(241,277)(242,276)(243,275)(244,274)
(245,273);;
s2 := ( 3,192)( 4,194)( 5,193)( 6,200)( 7,199)( 8,198)( 9,197)( 10,196)
( 11,195)( 12,218)( 13,217)( 14,216)( 15,215)( 16,214)( 17,213)( 18,212)
( 19,211)( 20,210)( 21,209)( 22,208)( 23,207)( 24,206)( 25,205)( 26,204)
( 27,203)( 28,202)( 29,201)( 30,165)( 31,167)( 32,166)( 33,173)( 34,172)
( 35,171)( 36,170)( 37,169)( 38,168)( 39,191)( 40,190)( 41,189)( 42,188)
( 43,187)( 44,186)( 45,185)( 46,184)( 47,183)( 48,182)( 49,181)( 50,180)
( 51,179)( 52,178)( 53,177)( 54,176)( 55,175)( 56,174)( 57,245)( 58,244)
( 59,243)( 60,242)( 61,241)( 62,240)( 63,239)( 64,238)( 65,237)( 66,236)
( 67,235)( 68,234)( 69,233)( 70,232)( 71,231)( 72,230)( 73,229)( 74,228)
( 75,227)( 76,226)( 77,225)( 78,224)( 79,223)( 80,222)( 81,221)( 82,220)
( 83,219)( 84,273)( 85,275)( 86,274)( 87,281)( 88,280)( 89,279)( 90,278)
( 91,277)( 92,276)( 93,299)( 94,298)( 95,297)( 96,296)( 97,295)( 98,294)
( 99,293)(100,292)(101,291)(102,290)(103,289)(104,288)(105,287)(106,286)
(107,285)(108,284)(109,283)(110,282)(111,246)(112,248)(113,247)(114,254)
(115,253)(116,252)(117,251)(118,250)(119,249)(120,272)(121,271)(122,270)
(123,269)(124,268)(125,267)(126,266)(127,265)(128,264)(129,263)(130,262)
(131,261)(132,260)(133,259)(134,258)(135,257)(136,256)(137,255)(138,326)
(139,325)(140,324)(141,323)(142,322)(143,321)(144,320)(145,319)(146,318)
(147,317)(148,316)(149,315)(150,314)(151,313)(152,312)(153,311)(154,310)
(155,309)(156,308)(157,307)(158,306)(159,305)(160,304)(161,303)(162,302)
(163,301)(164,300);;
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*s1*s0*s1, s0*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(326)!(1,2);
s1 := Sym(326)!( 4, 5)( 6, 11)( 7, 10)( 8, 9)( 12, 29)( 13, 28)( 14, 27)
( 15, 26)( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 30, 83)( 31, 82)
( 32, 81)( 33, 80)( 34, 79)( 35, 78)( 36, 77)( 37, 76)( 38, 75)( 39, 74)
( 40, 73)( 41, 72)( 42, 71)( 43, 70)( 44, 69)( 45, 68)( 46, 67)( 47, 66)
( 48, 65)( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 60)( 54, 59)( 55, 58)
( 56, 57)( 85, 86)( 87, 92)( 88, 91)( 89, 90)( 93,110)( 94,109)( 95,108)
( 96,107)( 97,106)( 98,105)( 99,104)(100,103)(101,102)(111,164)(112,163)
(113,162)(114,161)(115,160)(116,159)(117,158)(118,157)(119,156)(120,155)
(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)
(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)
(137,138)(165,246)(166,248)(167,247)(168,254)(169,253)(170,252)(171,251)
(172,250)(173,249)(174,272)(175,271)(176,270)(177,269)(178,268)(179,267)
(180,266)(181,265)(182,264)(183,263)(184,262)(185,261)(186,260)(187,259)
(188,258)(189,257)(190,256)(191,255)(192,326)(193,325)(194,324)(195,323)
(196,322)(197,321)(198,320)(199,319)(200,318)(201,317)(202,316)(203,315)
(204,314)(205,313)(206,312)(207,311)(208,310)(209,309)(210,308)(211,307)
(212,306)(213,305)(214,304)(215,303)(216,302)(217,301)(218,300)(219,299)
(220,298)(221,297)(222,296)(223,295)(224,294)(225,293)(226,292)(227,291)
(228,290)(229,289)(230,288)(231,287)(232,286)(233,285)(234,284)(235,283)
(236,282)(237,281)(238,280)(239,279)(240,278)(241,277)(242,276)(243,275)
(244,274)(245,273);
s2 := Sym(326)!( 3,192)( 4,194)( 5,193)( 6,200)( 7,199)( 8,198)( 9,197)
( 10,196)( 11,195)( 12,218)( 13,217)( 14,216)( 15,215)( 16,214)( 17,213)
( 18,212)( 19,211)( 20,210)( 21,209)( 22,208)( 23,207)( 24,206)( 25,205)
( 26,204)( 27,203)( 28,202)( 29,201)( 30,165)( 31,167)( 32,166)( 33,173)
( 34,172)( 35,171)( 36,170)( 37,169)( 38,168)( 39,191)( 40,190)( 41,189)
( 42,188)( 43,187)( 44,186)( 45,185)( 46,184)( 47,183)( 48,182)( 49,181)
( 50,180)( 51,179)( 52,178)( 53,177)( 54,176)( 55,175)( 56,174)( 57,245)
( 58,244)( 59,243)( 60,242)( 61,241)( 62,240)( 63,239)( 64,238)( 65,237)
( 66,236)( 67,235)( 68,234)( 69,233)( 70,232)( 71,231)( 72,230)( 73,229)
( 74,228)( 75,227)( 76,226)( 77,225)( 78,224)( 79,223)( 80,222)( 81,221)
( 82,220)( 83,219)( 84,273)( 85,275)( 86,274)( 87,281)( 88,280)( 89,279)
( 90,278)( 91,277)( 92,276)( 93,299)( 94,298)( 95,297)( 96,296)( 97,295)
( 98,294)( 99,293)(100,292)(101,291)(102,290)(103,289)(104,288)(105,287)
(106,286)(107,285)(108,284)(109,283)(110,282)(111,246)(112,248)(113,247)
(114,254)(115,253)(116,252)(117,251)(118,250)(119,249)(120,272)(121,271)
(122,270)(123,269)(124,268)(125,267)(126,266)(127,265)(128,264)(129,263)
(130,262)(131,261)(132,260)(133,259)(134,258)(135,257)(136,256)(137,255)
(138,326)(139,325)(140,324)(141,323)(142,322)(143,321)(144,320)(145,319)
(146,318)(147,317)(148,316)(149,315)(150,314)(151,313)(152,312)(153,311)
(154,310)(155,309)(156,308)(157,307)(158,306)(159,305)(160,304)(161,303)
(162,302)(163,301)(164,300);
poly := sub<Sym(326)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s1*s0*s1, s0*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope