Dic₁₃₈₄₃ #72290130

order	55372
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₄₃

Splittings

derived, Hall	C₁₀₉×C₁₂₇ ⋊ C₄
Sylow	C₁₂₇ ⋊ Dic₁₀₉
Sylow	C₁₀₉ ⋊ Dic₁₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₈₄₃
commutator, hypocenter	C₁₀₉×C₁₂₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₀₉×C₁₂₇	C₂
LCS	G ▷ C₁₀₉×C₁₂₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 240
transitive degree	μ_t(G) = 55372

A minimal faithful permutation representation — 2 generators on 240 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);(1,3,2,4)(6,113)(7,112)(8,111)(9,110)(10,109)(11,108)(12,107)(13,106)(14,105)(15,104)(16,103)(17,102)(18,101)(19,100)(20,99)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(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:

2^2,109,127|a^2=b,c^a=c^108,d^a=d^126

Conjugacy classes & character table

13846 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, 109, 127]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^108);;
gap> SetConjugate(coll, 4, 1, F.4^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13843 #72290130