Dic₁₆₂₇₉ #72301462

order	65116
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₂₇₉

Splittings

derived, Hall	C₇₃×C₂₂₃ ⋊ C₄
Sylow	C₂₂₃ ⋊ Dic₇₃
Sylow	C₇₃ ⋊ Dic₂₂₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₆₂₇₉
commutator, hypocenter	C₇₃×C₂₂₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇₃×C₂₂₃	C₂
LCS	G ▷ C₇₃×C₂₂₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 300
transitive degree	μ_t(G) = 65116

A minimal faithful permutation representation — 2 generators on 300 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);(1,3,2,4)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(79,300)(80,299)(81,298)(82,297)(83,296)(84,295)(85,294)(86,293)(87,292)(88,291)(89,290)(90,289)(91,288)(92,287)(93,286)(94,285)(95,284)(96,283)(97,282)(98,281)(99,280)(100,279)(101,278)(102,277)(103,276)(104,275)(105,274)(106,273)(107,272)(108,271)(109,270)(110,269)(111,268)(112,267)(113,266)(114,265)(115,264)(116,263)(117,262)(118,261)(119,260)(120,259)(121,258)(122,257)(123,256)(124,255)(125,254)(126,253)(127,252)(128,251)(129,250)(130,249)(131,248)(132,247)(133,246)(134,245)(135,244)(136,243)(137,242)(138,241)(139,240)(140,239)(141,238)(142,237)(143,236)(144,235)(145,234)(146,233)(147,232)(148,231)(149,230)(150,229)(151,228)(152,227)(153,226)(154,225)(155,224)(156,223)(157,222)(158,221)(159,220)(160,219)(161,218)(162,217)(163,216)(164,215)(165,214)(166,213)(167,212)(168,211)(169,210)(170,209)(171,208)(172,207)(173,206)(174,205)(175,204)(176,203)(177,202)(178,201)(179,200)(180,199)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)

Polycyclic presentation:

2^2,73,223|a^2=b,c^a=c^72,d^a=d^222

Conjugacy classes & character table

16282 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, 73, 223]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^72);;
gap> SetConjugate(coll, 4, 1, F.4^222);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16279 #72301462