Polytope of Type {312}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {312}*624
Also Known As : 312-gon, {312}. if this polytope has another name.
Group : SmallGroup(624,96)
Rank : 2
Schlafli Type : {312}
Number of vertices, edges, etc : 312, 312
Order of s0s1 : 312
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{312,2} of size 1248
Vertex Figure Of :
{2,312} of size 1248
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {156}*312
3-fold quotients : {104}*208
4-fold quotients : {78}*156
6-fold quotients : {52}*104
8-fold quotients : {39}*78
12-fold quotients : {26}*52
13-fold quotients : {24}*48
24-fold quotients : {13}*26
26-fold quotients : {12}*24
39-fold quotients : {8}*16
52-fold quotients : {6}*12
78-fold quotients : {4}*8
104-fold quotients : {3}*6
156-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {624}*1248
3-fold covers : {936}*1872
Permutation Representation (GAP) :
s0 := ( 2, 13)( 3, 12)( 4, 11)( 5, 10)( 6, 9)( 7, 8)( 14, 27)( 15, 39)
( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)( 23, 31)
( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)( 59, 73)
( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 79,118)( 80,130)
( 81,129)( 82,128)( 83,127)( 84,126)( 85,125)( 86,124)( 87,123)( 88,122)
( 89,121)( 90,120)( 91,119)( 92,144)( 93,156)( 94,155)( 95,154)( 96,153)
( 97,152)( 98,151)( 99,150)(100,149)(101,148)(102,147)(103,146)(104,145)
(105,131)(106,143)(107,142)(108,141)(109,140)(110,139)(111,138)(112,137)
(113,136)(114,135)(115,134)(116,133)(117,132)(157,235)(158,247)(159,246)
(160,245)(161,244)(162,243)(163,242)(164,241)(165,240)(166,239)(167,238)
(168,237)(169,236)(170,261)(171,273)(172,272)(173,271)(174,270)(175,269)
(176,268)(177,267)(178,266)(179,265)(180,264)(181,263)(182,262)(183,248)
(184,260)(185,259)(186,258)(187,257)(188,256)(189,255)(190,254)(191,253)
(192,252)(193,251)(194,250)(195,249)(196,274)(197,286)(198,285)(199,284)
(200,283)(201,282)(202,281)(203,280)(204,279)(205,278)(206,277)(207,276)
(208,275)(209,300)(210,312)(211,311)(212,310)(213,309)(214,308)(215,307)
(216,306)(217,305)(218,304)(219,303)(220,302)(221,301)(222,287)(223,299)
(224,298)(225,297)(226,296)(227,295)(228,294)(229,293)(230,292)(231,291)
(232,290)(233,289)(234,288);;
s1 := ( 1,171)( 2,170)( 3,182)( 4,181)( 5,180)( 6,179)( 7,178)( 8,177)
( 9,176)( 10,175)( 11,174)( 12,173)( 13,172)( 14,158)( 15,157)( 16,169)
( 17,168)( 18,167)( 19,166)( 20,165)( 21,164)( 22,163)( 23,162)( 24,161)
( 25,160)( 26,159)( 27,184)( 28,183)( 29,195)( 30,194)( 31,193)( 32,192)
( 33,191)( 34,190)( 35,189)( 36,188)( 37,187)( 38,186)( 39,185)( 40,210)
( 41,209)( 42,221)( 43,220)( 44,219)( 45,218)( 46,217)( 47,216)( 48,215)
( 49,214)( 50,213)( 51,212)( 52,211)( 53,197)( 54,196)( 55,208)( 56,207)
( 57,206)( 58,205)( 59,204)( 60,203)( 61,202)( 62,201)( 63,200)( 64,199)
( 65,198)( 66,223)( 67,222)( 68,234)( 69,233)( 70,232)( 71,231)( 72,230)
( 73,229)( 74,228)( 75,227)( 76,226)( 77,225)( 78,224)( 79,288)( 80,287)
( 81,299)( 82,298)( 83,297)( 84,296)( 85,295)( 86,294)( 87,293)( 88,292)
( 89,291)( 90,290)( 91,289)( 92,275)( 93,274)( 94,286)( 95,285)( 96,284)
( 97,283)( 98,282)( 99,281)(100,280)(101,279)(102,278)(103,277)(104,276)
(105,301)(106,300)(107,312)(108,311)(109,310)(110,309)(111,308)(112,307)
(113,306)(114,305)(115,304)(116,303)(117,302)(118,249)(119,248)(120,260)
(121,259)(122,258)(123,257)(124,256)(125,255)(126,254)(127,253)(128,252)
(129,251)(130,250)(131,236)(132,235)(133,247)(134,246)(135,245)(136,244)
(137,243)(138,242)(139,241)(140,240)(141,239)(142,238)(143,237)(144,262)
(145,261)(146,273)(147,272)(148,271)(149,270)(150,269)(151,268)(152,267)
(153,266)(154,265)(155,264)(156,263);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(312)!( 2, 13)( 3, 12)( 4, 11)( 5, 10)( 6, 9)( 7, 8)( 14, 27)
( 15, 39)( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)
( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)
( 45, 48)( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)
( 59, 73)( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 79,118)
( 80,130)( 81,129)( 82,128)( 83,127)( 84,126)( 85,125)( 86,124)( 87,123)
( 88,122)( 89,121)( 90,120)( 91,119)( 92,144)( 93,156)( 94,155)( 95,154)
( 96,153)( 97,152)( 98,151)( 99,150)(100,149)(101,148)(102,147)(103,146)
(104,145)(105,131)(106,143)(107,142)(108,141)(109,140)(110,139)(111,138)
(112,137)(113,136)(114,135)(115,134)(116,133)(117,132)(157,235)(158,247)
(159,246)(160,245)(161,244)(162,243)(163,242)(164,241)(165,240)(166,239)
(167,238)(168,237)(169,236)(170,261)(171,273)(172,272)(173,271)(174,270)
(175,269)(176,268)(177,267)(178,266)(179,265)(180,264)(181,263)(182,262)
(183,248)(184,260)(185,259)(186,258)(187,257)(188,256)(189,255)(190,254)
(191,253)(192,252)(193,251)(194,250)(195,249)(196,274)(197,286)(198,285)
(199,284)(200,283)(201,282)(202,281)(203,280)(204,279)(205,278)(206,277)
(207,276)(208,275)(209,300)(210,312)(211,311)(212,310)(213,309)(214,308)
(215,307)(216,306)(217,305)(218,304)(219,303)(220,302)(221,301)(222,287)
(223,299)(224,298)(225,297)(226,296)(227,295)(228,294)(229,293)(230,292)
(231,291)(232,290)(233,289)(234,288);
s1 := Sym(312)!( 1,171)( 2,170)( 3,182)( 4,181)( 5,180)( 6,179)( 7,178)
( 8,177)( 9,176)( 10,175)( 11,174)( 12,173)( 13,172)( 14,158)( 15,157)
( 16,169)( 17,168)( 18,167)( 19,166)( 20,165)( 21,164)( 22,163)( 23,162)
( 24,161)( 25,160)( 26,159)( 27,184)( 28,183)( 29,195)( 30,194)( 31,193)
( 32,192)( 33,191)( 34,190)( 35,189)( 36,188)( 37,187)( 38,186)( 39,185)
( 40,210)( 41,209)( 42,221)( 43,220)( 44,219)( 45,218)( 46,217)( 47,216)
( 48,215)( 49,214)( 50,213)( 51,212)( 52,211)( 53,197)( 54,196)( 55,208)
( 56,207)( 57,206)( 58,205)( 59,204)( 60,203)( 61,202)( 62,201)( 63,200)
( 64,199)( 65,198)( 66,223)( 67,222)( 68,234)( 69,233)( 70,232)( 71,231)
( 72,230)( 73,229)( 74,228)( 75,227)( 76,226)( 77,225)( 78,224)( 79,288)
( 80,287)( 81,299)( 82,298)( 83,297)( 84,296)( 85,295)( 86,294)( 87,293)
( 88,292)( 89,291)( 90,290)( 91,289)( 92,275)( 93,274)( 94,286)( 95,285)
( 96,284)( 97,283)( 98,282)( 99,281)(100,280)(101,279)(102,278)(103,277)
(104,276)(105,301)(106,300)(107,312)(108,311)(109,310)(110,309)(111,308)
(112,307)(113,306)(114,305)(115,304)(116,303)(117,302)(118,249)(119,248)
(120,260)(121,259)(122,258)(123,257)(124,256)(125,255)(126,254)(127,253)
(128,252)(129,251)(130,250)(131,236)(132,235)(133,247)(134,246)(135,245)
(136,244)(137,243)(138,242)(139,241)(140,240)(141,239)(142,238)(143,237)
(144,262)(145,261)(146,273)(147,272)(148,271)(149,270)(150,269)(151,268)
(152,267)(153,266)(154,265)(155,264)(156,263);
poly := sub<Sym(312)|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 >;
References : None.
to this polytope