Dic₁₂₇₅₁ #72288147

order	51004
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₇₅₁

Splittings

derived, Hall	C₄₁×C₃₁₁ ⋊ C₄
Sylow	C₃₁₁ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₃₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₂₇₅₁
commutator, hypocenter	C₄₁×C₃₁₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₁×C₃₁₁	C₂
LCS	G ▷ C₄₁×C₃₁₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 356
transitive degree	μ_t(G) = 51004

A minimal faithful permutation representation — 2 generators on 356 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);(1,3,2,4)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(47,356)(48,355)(49,354)(50,353)(51,352)(52,351)(53,350)(54,349)(55,348)(56,347)(57,346)(58,345)(59,344)(60,343)(61,342)(62,341)(63,340)(64,339)(65,338)(66,337)(67,336)(68,335)(69,334)(70,333)(71,332)(72,331)(73,330)(74,329)(75,328)(76,327)(77,326)(78,325)(79,324)(80,323)(81,322)(82,321)(83,320)(84,319)(85,318)(86,317)(87,316)(88,315)(89,314)(90,313)(91,312)(92,311)(93,310)(94,309)(95,308)(96,307)(97,306)(98,305)(99,304)(100,303)(101,302)(102,301)(103,300)(104,299)(105,298)(106,297)(107,296)(108,295)(109,294)(110,293)(111,292)(112,291)(113,290)(114,289)(115,288)(116,287)(117,286)(118,285)(119,284)(120,283)(121,282)(122,281)(123,280)(124,279)(125,278)(126,277)(127,276)(128,275)(129,274)(130,273)(131,272)(132,271)(133,270)(134,269)(135,268)(136,267)(137,266)(138,265)(139,264)(140,263)(141,262)(142,261)(143,260)(144,259)(145,258)(146,257)(147,256)(148,255)(149,254)(150,253)(151,252)(152,251)(153,250)(154,249)(155,248)(156,247)(157,246)(158,245)(159,244)(160,243)(161,242)(162,241)(163,240)(164,239)(165,238)(166,237)(167,236)(168,235)(169,234)(170,233)(171,232)(172,231)(173,230)(174,229)(175,228)(176,227)(177,226)(178,225)(179,224)(180,223)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,210)(194,209)(195,208)(196,207)(197,206)(198,205)(199,204)(200,203)(201,202)

Polycyclic presentation:

2^2,41,311|a^2=b,c^a=c^40,d^a=d^310

Conjugacy classes & character table

12754 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, 41, 311]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^40);;
gap> SetConjugate(coll, 4, 1, F.4^310);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12751 #72288147