Dic₁₀₄₀₆ #72274443

order	41624
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₄₀₆

Splittings

Hall	C₄₃×C₁₂₁ ⋊ Q₈
Sylow	C₁₂₁ ⋊ Dic₈₆
Sylow	C₄₃ ⋊ Dic₂₄₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅₂₀₃
commutator, cosocle	C₂×C₄₃×C₁₂₁	C₂²
Frattini	C₂×C₁₁	C₂×D₄₇₃
Fitting	C₄×C₄₃×C₁₂₁	C₂
socle, prime radical	C₂×C₁₁×C₄₃	C₂×D₁₁
hypercenter	C₄	D₅₂₀₃
hypocenter	C₄₃×C₁₂₁	Q₈
LCS	G ▷ C₂×C₄₃×C₁₂₁ ▷ C₄₃×C₁₂₁
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 172
transitive degree	μ_t(G) = 41624

A minimal faithful permutation representation — 2 generators on 172 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,129)(11,128)(12,127)(13,126)(14,125)(15,124)(16,123)(17,122)(18,121)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)

Polycyclic presentation:

2^3,11,43,11|a^2=c,b^2=c,d^11=f,b^a=bc,d^a=d^10f^10,e^a=e^42,f^a=f^10

Conjugacy classes & character table

10409 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", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 11, 43, 11]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^10*F.6^10);;
gap> SetConjugate(coll, 5, 1, F.5^42);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic10406 #72274443