D₂₃₇₆₄ #72283129

order	47528
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₃₇₆₄

Splittings

Fitting	C₄×C₁₃×C₄₅₇ ⋊ C₂
Hall	C₁₃×C₄₅₇ ⋊ D₄
Sylow	C₄₅₇ ⋊ D₅₂
Sylow	C₁₃ ⋊ D₁₈₂₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₉₄₁
commutator, socle, cosocle	C₂×C₁₃×C₄₅₇	C₂²
Fitting	C₄×C₁₃×C₄₅₇	C₂
hypercenter	C₄	D₅₉₄₁
hypocenter	C₁₃×C₄₅₇	D₄
LCS	G ▷ C₂×C₁₃×C₄₅₇ ▷ C₁₃×C₄₅₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 474
transitive degree	μ_t(G) = 23764

A minimal faithful permutation representation — 2 generators on 474 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474);(2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,474)(20,473)(21,472)(22,471)(23,470)(24,469)(25,468)(26,467)(27,466)(28,465)(29,464)(30,463)(31,462)(32,461)(33,460)(34,459)(35,458)(36,457)(37,456)(38,455)(39,454)(40,453)(41,452)(42,451)(43,450)(44,449)(45,448)(46,447)(47,446)(48,445)(49,444)(50,443)(51,442)(52,441)(53,440)(54,439)(55,438)(56,437)(57,436)(58,435)(59,434)(60,433)(61,432)(62,431)(63,430)(64,429)(65,428)(66,427)(67,426)(68,425)(69,424)(70,423)(71,422)(72,421)(73,420)(74,419)(75,418)(76,417)(77,416)(78,415)(79,414)(80,413)(81,412)(82,411)(83,410)(84,409)(85,408)(86,407)(87,406)(88,405)(89,404)(90,403)(91,402)(92,401)(93,400)(94,399)(95,398)(96,397)(97,396)(98,395)(99,394)(100,393)(101,392)(102,391)(103,390)(104,389)(105,388)(106,387)(107,386)(108,385)(109,384)(110,383)(111,382)(112,381)(113,380)(114,379)(115,378)(116,377)(117,376)(118,375)(119,374)(120,373)(121,372)(122,371)(123,370)(124,369)(125,368)(126,367)(127,366)(128,365)(129,364)(130,363)(131,362)(132,361)(133,360)(134,359)(135,358)(136,357)(137,356)(138,355)(139,354)(140,353)(141,352)(142,351)(143,350)(144,349)(145,348)(146,347)(147,346)(148,345)(149,344)(150,343)(151,342)(152,341)(153,340)(154,339)(155,338)(156,337)(157,336)(158,335)(159,334)(160,333)(161,332)(162,331)(163,330)(164,329)(165,328)(166,327)(167,326)(168,325)(169,324)(170,323)(171,322)(172,321)(173,320)(174,319)(175,318)(176,317)(177,316)(178,315)(179,314)(180,313)(181,312)(182,311)(183,310)(184,309)(185,308)(186,307)(187,306)(188,305)(189,304)(190,303)(191,302)(192,301)(193,300)(194,299)(195,298)(196,297)(197,296)(198,295)(199,294)(200,293)(201,292)(202,291)(203,290)(204,289)(205,288)(206,287)(207,286)(208,285)(209,284)(210,283)(211,282)(212,281)(213,280)(214,279)(215,278)(216,277)(217,276)(218,275)(219,274)(220,273)(221,272)(222,271)(223,270)(224,269)(225,268)(226,267)(227,266)(228,265)(229,264)(230,263)(231,262)(232,261)(233,260)(234,259)(235,258)(236,257)(237,256)(238,255)(239,254)(240,253)(241,252)(242,251)(243,250)(244,249)(245,248)(246,247)

Polycyclic presentation:

2^3,13,457|b^2=c,b^a=bc,d^a=d^12,e^a=e^456

Conjugacy classes & character table

11885 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 13, 457]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^456);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D23764 #72283129