Dic₁₄₃₁₇ #72290922

order	57268
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₁₇

Splittings

derived, Hall	C₁₀₃×C₁₃₉ ⋊ C₄
Sylow	C₁₃₉ ⋊ Dic₁₀₃
Sylow	C₁₀₃ ⋊ Dic₁₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₃₁₇
commutator, hypocenter	C₁₀₃×C₁₃₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₀₃×C₁₃₉	C₂
LCS	G ▷ C₁₀₃×C₁₃₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 246
transitive degree	μ_t(G) = 57268

A minimal faithful permutation representation — 2 generators on 246 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);(1,3,2,4)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(109,246)(110,245)(111,244)(112,243)(113,242)(114,241)(115,240)(116,239)(117,238)(118,237)(119,236)(120,235)(121,234)(122,233)(123,232)(124,231)(125,230)(126,229)(127,228)(128,227)(129,226)(130,225)(131,224)(132,223)(133,222)(134,221)(135,220)(136,219)(137,218)(138,217)(139,216)(140,215)(141,214)(142,213)(143,212)(144,211)(145,210)(146,209)(147,208)(148,207)(149,206)(150,205)(151,204)(152,203)(153,202)(154,201)(155,200)(156,199)(157,198)(158,197)(159,196)(160,195)(161,194)(162,193)(163,192)(164,191)(165,190)(166,189)(167,188)(168,187)(169,186)(170,185)(171,184)(172,183)(173,182)(174,181)(175,180)(176,179)(177,178)

Polycyclic presentation:

2^2,103,139|a^2=b,c^a=c^102,d^a=d^138

Conjugacy classes & character table

14320 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, 103, 139]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^102);;
gap> SetConjugate(coll, 4, 1, F.4^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14317 #72290922