D₃₂₇₅₆ #72304518

order	65512
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₇₅₆

Splittings

Fitting	C₄×C₁₉×C₄₃₁ ⋊ C₂
Hall	C₁₉×C₄₃₁ ⋊ D₄
Sylow	C₄₃₁ ⋊ D₇₆
Sylow	C₁₉ ⋊ D₁₇₂₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₈₁₈₉
commutator, socle, cosocle	C₂×C₁₉×C₄₃₁	C₂²
Fitting	C₄×C₁₉×C₄₃₁	C₂
hypercenter	C₄	D₈₁₈₉
hypocenter	C₁₉×C₄₃₁	D₄
LCS	G ▷ C₂×C₁₉×C₄₃₁ ▷ C₁₉×C₄₃₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 454
transitive degree	μ_t(G) = 32756

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

Polycyclic presentation:

2^3,19,431|b^2=c,b^a=bc,d^a=d^18,e^a=e^430

Conjugacy classes & character table

16381 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, 19, 431]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^430);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32756 #72304518