Dic₁₄₁₄₁ #72290609

order	56564
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₁₄₁

Splittings

derived, Hall	C₇₉×C₁₇₉ ⋊ C₄
Sylow	C₁₇₉ ⋊ Dic₇₉
Sylow	C₇₉ ⋊ Dic₁₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₁₄₁
commutator, hypocenter	C₇₉×C₁₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇₉×C₁₇₉	C₂
LCS	G ▷ C₇₉×C₁₇₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 262
transitive degree	μ_t(G) = 56564

A minimal faithful permutation representation — 2 generators on 262 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);(1,3,2,4)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(85,262)(86,261)(87,260)(88,259)(89,258)(90,257)(91,256)(92,255)(93,254)(94,253)(95,252)(96,251)(97,250)(98,249)(99,248)(100,247)(101,246)(102,245)(103,244)(104,243)(105,242)(106,241)(107,240)(108,239)(109,238)(110,237)(111,236)(112,235)(113,234)(114,233)(115,232)(116,231)(117,230)(118,229)(119,228)(120,227)(121,226)(122,225)(123,224)(124,223)(125,222)(126,221)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)(146,201)(147,200)(148,199)(149,198)(150,197)(151,196)(152,195)(153,194)(154,193)(155,192)(156,191)(157,190)(158,189)(159,188)(160,187)(161,186)(162,185)(163,184)(164,183)(165,182)(166,181)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)

Polycyclic presentation:

2^2,79,179|a^2=b,c^a=c^78,d^a=d^178

Conjugacy classes & character table

14144 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, 79, 179]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^78);;
gap> SetConjugate(coll, 4, 1, F.4^178);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14141 #72290609