Dic₁₄₂₁₉ #72290733

order	56876
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₂₁₉

Splittings

derived, Hall	C₅₉×C₂₄₁ ⋊ C₄
Sylow	C₂₄₁ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₂₄₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₂₁₉
commutator, hypocenter	C₅₉×C₂₄₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅₉×C₂₄₁	C₂
LCS	G ▷ C₅₉×C₂₄₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 304
transitive degree	μ_t(G) = 56876

A minimal faithful permutation representation — 2 generators on 304 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);(1,3,2,4)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(65,304)(66,303)(67,302)(68,301)(69,300)(70,299)(71,298)(72,297)(73,296)(74,295)(75,294)(76,293)(77,292)(78,291)(79,290)(80,289)(81,288)(82,287)(83,286)(84,285)(85,284)(86,283)(87,282)(88,281)(89,280)(90,279)(91,278)(92,277)(93,276)(94,275)(95,274)(96,273)(97,272)(98,271)(99,270)(100,269)(101,268)(102,267)(103,266)(104,265)(105,264)(106,263)(107,262)(108,261)(109,260)(110,259)(111,258)(112,257)(113,256)(114,255)(115,254)(116,253)(117,252)(118,251)(119,250)(120,249)(121,248)(122,247)(123,246)(124,245)(125,244)(126,243)(127,242)(128,241)(129,240)(130,239)(131,238)(132,237)(133,236)(134,235)(135,234)(136,233)(137,232)(138,231)(139,230)(140,229)(141,228)(142,227)(143,226)(144,225)(145,224)(146,223)(147,222)(148,221)(149,220)(150,219)(151,218)(152,217)(153,216)(154,215)(155,214)(156,213)(157,212)(158,211)(159,210)(160,209)(161,208)(162,207)(163,206)(164,205)(165,204)(166,203)(167,202)(168,201)(169,200)(170,199)(171,198)(172,197)(173,196)(174,195)(175,194)(176,193)(177,192)(178,191)(179,190)(180,189)(181,188)(182,187)(183,186)(184,185)

Polycyclic presentation:

2^2,59,241|a^2=b,c^a=c^58,d^a=d^240

Conjugacy classes & character table

14222 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, 2, 59, 241]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^58);;
gap> SetConjugate(coll, 4, 1, F.4^240);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14219 #72290733