Polytope of Type {324}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {324}*648
Also Known As : 324-gon, {324}. if this polytope has another name.
Group : SmallGroup(648,6)
Rank : 2
Schlafli Type : {324}
Number of vertices, edges, etc : 324, 324
Order of s0s1 : 324
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{324,2} of size 1296
Vertex Figure Of :
{2,324} of size 1296
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {162}*324
3-fold quotients : {108}*216
4-fold quotients : {81}*162
6-fold quotients : {54}*108
9-fold quotients : {36}*72
12-fold quotients : {27}*54
18-fold quotients : {18}*36
27-fold quotients : {12}*24
36-fold quotients : {9}*18
54-fold quotients : {6}*12
81-fold quotients : {4}*8
108-fold quotients : {3}*6
162-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {648}*1296
3-fold covers : {972}*1944
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 28, 81)( 29, 80)( 30, 79)
( 31, 78)( 32, 77)( 33, 76)( 34, 75)( 35, 74)( 36, 73)( 37, 72)( 38, 71)
( 39, 70)( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)( 46, 63)
( 47, 62)( 48, 61)( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)( 54, 55)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)
( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(109,162)(110,161)(111,160)
(112,159)(113,158)(114,157)(115,156)(116,155)(117,154)(118,153)(119,152)
(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,144)
(128,143)(129,142)(130,141)(131,140)(132,139)(133,138)(134,137)(135,136)
(163,244)(164,246)(165,245)(166,252)(167,251)(168,250)(169,249)(170,248)
(171,247)(172,270)(173,269)(174,268)(175,267)(176,266)(177,265)(178,264)
(179,263)(180,262)(181,261)(182,260)(183,259)(184,258)(185,257)(186,256)
(187,255)(188,254)(189,253)(190,324)(191,323)(192,322)(193,321)(194,320)
(195,319)(196,318)(197,317)(198,316)(199,315)(200,314)(201,313)(202,312)
(203,311)(204,310)(205,309)(206,308)(207,307)(208,306)(209,305)(210,304)
(211,303)(212,302)(213,301)(214,300)(215,299)(216,298)(217,297)(218,296)
(219,295)(220,294)(221,293)(222,292)(223,291)(224,290)(225,289)(226,288)
(227,287)(228,286)(229,285)(230,284)(231,283)(232,282)(233,281)(234,280)
(235,279)(236,278)(237,277)(238,276)(239,275)(240,274)(241,273)(242,272)
(243,271);;
s1 := ( 1,190)( 2,192)( 3,191)( 4,198)( 5,197)( 6,196)( 7,195)( 8,194)
( 9,193)( 10,216)( 11,215)( 12,214)( 13,213)( 14,212)( 15,211)( 16,210)
( 17,209)( 18,208)( 19,207)( 20,206)( 21,205)( 22,204)( 23,203)( 24,202)
( 25,201)( 26,200)( 27,199)( 28,163)( 29,165)( 30,164)( 31,171)( 32,170)
( 33,169)( 34,168)( 35,167)( 36,166)( 37,189)( 38,188)( 39,187)( 40,186)
( 41,185)( 42,184)( 43,183)( 44,182)( 45,181)( 46,180)( 47,179)( 48,178)
( 49,177)( 50,176)( 51,175)( 52,174)( 53,173)( 54,172)( 55,243)( 56,242)
( 57,241)( 58,240)( 59,239)( 60,238)( 61,237)( 62,236)( 63,235)( 64,234)
( 65,233)( 66,232)( 67,231)( 68,230)( 69,229)( 70,228)( 71,227)( 72,226)
( 73,225)( 74,224)( 75,223)( 76,222)( 77,221)( 78,220)( 79,219)( 80,218)
( 81,217)( 82,271)( 83,273)( 84,272)( 85,279)( 86,278)( 87,277)( 88,276)
( 89,275)( 90,274)( 91,297)( 92,296)( 93,295)( 94,294)( 95,293)( 96,292)
( 97,291)( 98,290)( 99,289)(100,288)(101,287)(102,286)(103,285)(104,284)
(105,283)(106,282)(107,281)(108,280)(109,244)(110,246)(111,245)(112,252)
(113,251)(114,250)(115,249)(116,248)(117,247)(118,270)(119,269)(120,268)
(121,267)(122,266)(123,265)(124,264)(125,263)(126,262)(127,261)(128,260)
(129,259)(130,258)(131,257)(132,256)(133,255)(134,254)(135,253)(136,324)
(137,323)(138,322)(139,321)(140,320)(141,319)(142,318)(143,317)(144,316)
(145,315)(146,314)(147,313)(148,312)(149,311)(150,310)(151,309)(152,308)
(153,307)(154,306)(155,305)(156,304)(157,303)(158,302)(159,301)(160,300)
(161,299)(162,298);;
poly := Group([s0,s1]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;; s1 := F.2;;
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 28, 81)( 29, 80)
( 30, 79)( 31, 78)( 32, 77)( 33, 76)( 34, 75)( 35, 74)( 36, 73)( 37, 72)
( 38, 71)( 39, 70)( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)
( 46, 63)( 47, 62)( 48, 61)( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)
( 54, 55)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)
( 94,105)( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(109,162)(110,161)
(111,160)(112,159)(113,158)(114,157)(115,156)(116,155)(117,154)(118,153)
(119,152)(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)
(127,144)(128,143)(129,142)(130,141)(131,140)(132,139)(133,138)(134,137)
(135,136)(163,244)(164,246)(165,245)(166,252)(167,251)(168,250)(169,249)
(170,248)(171,247)(172,270)(173,269)(174,268)(175,267)(176,266)(177,265)
(178,264)(179,263)(180,262)(181,261)(182,260)(183,259)(184,258)(185,257)
(186,256)(187,255)(188,254)(189,253)(190,324)(191,323)(192,322)(193,321)
(194,320)(195,319)(196,318)(197,317)(198,316)(199,315)(200,314)(201,313)
(202,312)(203,311)(204,310)(205,309)(206,308)(207,307)(208,306)(209,305)
(210,304)(211,303)(212,302)(213,301)(214,300)(215,299)(216,298)(217,297)
(218,296)(219,295)(220,294)(221,293)(222,292)(223,291)(224,290)(225,289)
(226,288)(227,287)(228,286)(229,285)(230,284)(231,283)(232,282)(233,281)
(234,280)(235,279)(236,278)(237,277)(238,276)(239,275)(240,274)(241,273)
(242,272)(243,271);
s1 := Sym(324)!( 1,190)( 2,192)( 3,191)( 4,198)( 5,197)( 6,196)( 7,195)
( 8,194)( 9,193)( 10,216)( 11,215)( 12,214)( 13,213)( 14,212)( 15,211)
( 16,210)( 17,209)( 18,208)( 19,207)( 20,206)( 21,205)( 22,204)( 23,203)
( 24,202)( 25,201)( 26,200)( 27,199)( 28,163)( 29,165)( 30,164)( 31,171)
( 32,170)( 33,169)( 34,168)( 35,167)( 36,166)( 37,189)( 38,188)( 39,187)
( 40,186)( 41,185)( 42,184)( 43,183)( 44,182)( 45,181)( 46,180)( 47,179)
( 48,178)( 49,177)( 50,176)( 51,175)( 52,174)( 53,173)( 54,172)( 55,243)
( 56,242)( 57,241)( 58,240)( 59,239)( 60,238)( 61,237)( 62,236)( 63,235)
( 64,234)( 65,233)( 66,232)( 67,231)( 68,230)( 69,229)( 70,228)( 71,227)
( 72,226)( 73,225)( 74,224)( 75,223)( 76,222)( 77,221)( 78,220)( 79,219)
( 80,218)( 81,217)( 82,271)( 83,273)( 84,272)( 85,279)( 86,278)( 87,277)
( 88,276)( 89,275)( 90,274)( 91,297)( 92,296)( 93,295)( 94,294)( 95,293)
( 96,292)( 97,291)( 98,290)( 99,289)(100,288)(101,287)(102,286)(103,285)
(104,284)(105,283)(106,282)(107,281)(108,280)(109,244)(110,246)(111,245)
(112,252)(113,251)(114,250)(115,249)(116,248)(117,247)(118,270)(119,269)
(120,268)(121,267)(122,266)(123,265)(124,264)(125,263)(126,262)(127,261)
(128,260)(129,259)(130,258)(131,257)(132,256)(133,255)(134,254)(135,253)
(136,324)(137,323)(138,322)(139,321)(140,320)(141,319)(142,318)(143,317)
(144,316)(145,315)(146,314)(147,313)(148,312)(149,311)(150,310)(151,309)
(152,308)(153,307)(154,306)(155,305)(156,304)(157,303)(158,302)(159,301)
(160,300)(161,299)(162,298);
poly := sub<Sym(324)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope