Dic₁₂₈₂₇ #72288264

order	51308
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₈₂₇

Splittings

derived, Hall	C₁₀₁×C₁₂₇ ⋊ C₄
Sylow	C₁₂₇ ⋊ Dic₁₀₁
Sylow	C₁₀₁ ⋊ Dic₁₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₂₈₂₇
commutator, hypocenter	C₁₀₁×C₁₂₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₀₁×C₁₂₇	C₂
LCS	G ▷ C₁₀₁×C₁₂₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 232
transitive degree	μ_t(G) = 51308

A minimal faithful permutation representation — 2 generators on 232 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);(1,3,2,4)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(107,232)(108,231)(109,230)(110,229)(111,228)(112,227)(113,226)(114,225)(115,224)(116,223)(117,222)(118,221)(119,220)(120,219)(121,218)(122,217)(123,216)(124,215)(125,214)(126,213)(127,212)(128,211)(129,210)(130,209)(131,208)(132,207)(133,206)(134,205)(135,204)(136,203)(137,202)(138,201)(139,200)(140,199)(141,198)(142,197)(143,196)(144,195)(145,194)(146,193)(147,192)(148,191)(149,190)(150,189)(151,188)(152,187)(153,186)(154,185)(155,184)(156,183)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)

Polycyclic presentation:

2^2,101,127|a^2=b,c^a=c^100,d^a=d^126

Conjugacy classes & character table

12830 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, 101, 127]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^100);;
gap> SetConjugate(coll, 4, 1, F.4^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12827 #72288264