Dic₁₄₅₇₉ #72291389

order	58316
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₅₇₉

Splittings

derived, Hall	C₆₁×C₂₃₉ ⋊ C₄
Sylow	C₂₃₉ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₂₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₅₇₉
commutator, hypocenter	C₆₁×C₂₃₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₆₁×C₂₃₉	C₂
LCS	G ▷ C₆₁×C₂₃₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 304
transitive degree	μ_t(G) = 58316

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)(6,65)(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)(67,304)(68,303)(69,302)(70,301)(71,300)(72,299)(73,298)(74,297)(75,296)(76,295)(77,294)(78,293)(79,292)(80,291)(81,290)(82,289)(83,288)(84,287)(85,286)(86,285)(87,284)(88,283)(89,282)(90,281)(91,280)(92,279)(93,278)(94,277)(95,276)(96,275)(97,274)(98,273)(99,272)(100,271)(101,270)(102,269)(103,268)(104,267)(105,266)(106,265)(107,264)(108,263)(109,262)(110,261)(111,260)(112,259)(113,258)(114,257)(115,256)(116,255)(117,254)(118,253)(119,252)(120,251)(121,250)(122,249)(123,248)(124,247)(125,246)(126,245)(127,244)(128,243)(129,242)(130,241)(131,240)(132,239)(133,238)(134,237)(135,236)(136,235)(137,234)(138,233)(139,232)(140,231)(141,230)(142,229)(143,228)(144,227)(145,226)(146,225)(147,224)(148,223)(149,222)(150,221)(151,220)(152,219)(153,218)(154,217)(155,216)(156,215)(157,214)(158,213)(159,212)(160,211)(161,210)(162,209)(163,208)(164,207)(165,206)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)(181,190)(182,189)(183,188)(184,187)(185,186)

Polycyclic presentation:

2^2,61,239|a^2=b,c^a=c^60,d^a=d^238

Conjugacy classes & character table

14582 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, 239]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^60);;
gap> SetConjugate(coll, 4, 1, F.4^238);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14579 #72291389