D₂₈₉₁₉ #72292525

order	57838
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₈₉₁₉

Splittings

Fitting, derived, Hall	C₁₂₁×C₂₃₉ ⋊ C₂
Sylow	C₂₃₉ ⋊ D₁₂₁
Sylow	C₁₂₁ ⋊ D₂₃₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₂₁×C₂₃₉	C₂
Frattini	C₁₁	D₂₆₂₉
socle	C₁₁×C₂₃₉	D₁₁
LCS	G ▷ C₁₂₁×C₂₃₉
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 360
transitive degree	μ_t(G) = 28919

A minimal faithful permutation representation — 2 generators on 360 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);(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(123,360)(124,359)(125,358)(126,357)(127,356)(128,355)(129,354)(130,353)(131,352)(132,351)(133,350)(134,349)(135,348)(136,347)(137,346)(138,345)(139,344)(140,343)(141,342)(142,341)(143,340)(144,339)(145,338)(146,337)(147,336)(148,335)(149,334)(150,333)(151,332)(152,331)(153,330)(154,329)(155,328)(156,327)(157,326)(158,325)(159,324)(160,323)(161,322)(162,321)(163,320)(164,319)(165,318)(166,317)(167,316)(168,315)(169,314)(170,313)(171,312)(172,311)(173,310)(174,309)(175,308)(176,307)(177,306)(178,305)(179,304)(180,303)(181,302)(182,301)(183,300)(184,299)(185,298)(186,297)(187,296)(188,295)(189,294)(190,293)(191,292)(192,291)(193,290)(194,289)(195,288)(196,287)(197,286)(198,285)(199,284)(200,283)(201,282)(202,281)(203,280)(204,279)(205,278)(206,277)(207,276)(208,275)(209,274)(210,273)(211,272)(212,271)(213,270)(214,269)(215,268)(216,267)(217,266)(218,265)(219,264)(220,263)(221,262)(222,261)(223,260)(224,259)(225,258)(226,257)(227,256)(228,255)(229,254)(230,253)(231,252)(232,251)(233,250)(234,249)(235,248)(236,247)(237,246)(238,245)(239,244)(240,243)(241,242)

Polycyclic presentation:

2,11,239,11|b^11=d,b^a=b^10d^10,c^a=c^238,d^a=d^10

Conjugacy classes & character table

14461 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");;
gap> coll := SingleCollector(F, [2, 11, 239, 11]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2^10*F.4^10);;
gap> SetConjugate(coll, 3, 1, F.3^238);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28919 #72292525