Dic₁₃₁₈₉ #72288835

order	52756
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₁₈₉

Splittings

derived, Hall	C₁₀₉×C₁₂₁ ⋊ C₄
Sylow	C₁₂₁ ⋊ Dic₁₀₉
Sylow	C₁₀₉ ⋊ Dic₁₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₁₈₉
commutator, hypocenter	C₁₀₉×C₁₂₁	C₄
Frattini	C₂×C₁₁	D₁₁₉₉
Fitting, cosocle	C₂×C₁₀₉×C₁₂₁	C₂
socle, prime radical	C₂×C₁₁×C₁₀₉	D₁₁
LCS	G ▷ C₁₀₉×C₁₂₁
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 234
transitive degree	μ_t(G) = 52756

A minimal faithful permutation representation — 2 generators on 234 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);(1,3,2,4)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(127,234)(128,233)(129,232)(130,231)(131,230)(132,229)(133,228)(134,227)(135,226)(136,225)(137,224)(138,223)(139,222)(140,221)(141,220)(142,219)(143,218)(144,217)(145,216)(146,215)(147,214)(148,213)(149,212)(150,211)(151,210)(152,209)(153,208)(154,207)(155,206)(156,205)(157,204)(158,203)(159,202)(160,201)(161,200)(162,199)(163,198)(164,197)(165,196)(166,195)(167,194)(168,193)(169,192)(170,191)(171,190)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)

Polycyclic presentation:

2^2,11,109,11|a^2=b,c^11=e,c^a=c^10e^10,d^a=d^108,e^a=e^10

Conjugacy classes & character table

13192 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", "e");;
gap> coll := SingleCollector(F, [2, 2, 11, 109, 11]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 3, 1, F.3^10*F.5^10);;
gap> SetConjugate(coll, 4, 1, F.4^108);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13189 #72288835