Dic₁₄₈₇₃ #72292191

order	59492
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₈₇₃

Splittings

derived, Hall	C₁₀₇×C₁₃₉ ⋊ C₄
Sylow	C₁₃₉ ⋊ Dic₁₀₇
Sylow	C₁₀₇ ⋊ Dic₁₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₈₇₃
commutator, hypocenter	C₁₀₇×C₁₃₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₀₇×C₁₃₉	C₂
LCS	G ▷ C₁₀₇×C₁₃₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 250
transitive degree	μ_t(G) = 59492

A minimal faithful permutation representation — 2 generators on 250 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);(1,3,2,4)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(113,250)(114,249)(115,248)(116,247)(117,246)(118,245)(119,244)(120,243)(121,242)(122,241)(123,240)(124,239)(125,238)(126,237)(127,236)(128,235)(129,234)(130,233)(131,232)(132,231)(133,230)(134,229)(135,228)(136,227)(137,226)(138,225)(139,224)(140,223)(141,222)(142,221)(143,220)(144,219)(145,218)(146,217)(147,216)(148,215)(149,214)(150,213)(151,212)(152,211)(153,210)(154,209)(155,208)(156,207)(157,206)(158,205)(159,204)(160,203)(161,202)(162,201)(163,200)(164,199)(165,198)(166,197)(167,196)(168,195)(169,194)(170,193)(171,192)(172,191)(173,190)(174,189)(175,188)(176,187)(177,186)(178,185)(179,184)(180,183)(181,182)

Polycyclic presentation:

2^2,107,139|a^2=b,c^a=c^106,d^a=d^138

Conjugacy classes & character table

14876 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, 107, 139]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^106);;
gap> SetConjugate(coll, 4, 1, F.4^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14873 #72292191