Dic₁₃₉₆₉ #72290331

order	55876
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₉₆₉

Splittings

derived, Hall	C₆₁×C₂₂₉ ⋊ C₄
Sylow	C₂₂₉ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₂₂₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₉₆₉
commutator, hypocenter	C₆₁×C₂₂₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₆₁×C₂₂₉	C₂
LCS	G ▷ C₆₁×C₂₂₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 294
transitive degree	μ_t(G) = 55876

A minimal faithful permutation representation — 2 generators on 294 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);(1,3,2,4)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(67,294)(68,293)(69,292)(70,291)(71,290)(72,289)(73,288)(74,287)(75,286)(76,285)(77,284)(78,283)(79,282)(80,281)(81,280)(82,279)(83,278)(84,277)(85,276)(86,275)(87,274)(88,273)(89,272)(90,271)(91,270)(92,269)(93,268)(94,267)(95,266)(96,265)(97,264)(98,263)(99,262)(100,261)(101,260)(102,259)(103,258)(104,257)(105,256)(106,255)(107,254)(108,253)(109,252)(110,251)(111,250)(112,249)(113,248)(114,247)(115,246)(116,245)(117,244)(118,243)(119,242)(120,241)(121,240)(122,239)(123,238)(124,237)(125,236)(126,235)(127,234)(128,233)(129,232)(130,231)(131,230)(132,229)(133,228)(134,227)(135,226)(136,225)(137,224)(138,223)(139,222)(140,221)(141,220)(142,219)(143,218)(144,217)(145,216)(146,215)(147,214)(148,213)(149,212)(150,211)(151,210)(152,209)(153,208)(154,207)(155,206)(156,205)(157,204)(158,203)(159,202)(160,201)(161,200)(162,199)(163,198)(164,197)(165,196)(166,195)(167,194)(168,193)(169,192)(170,191)(171,190)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)

Polycyclic presentation:

2^2,61,229|a^2=b,c^a=c^60,d^a=d^228

Conjugacy classes & character table

13972 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, 61, 229]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^60);;
gap> SetConjugate(coll, 4, 1, F.4^228);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13969 #72290331