Dic₄₀₀₉ #30630975

order	16036
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₀₀₉
unambiguous semidirect product	C₁₉⋊Dic₂₁₁
unambiguous semidirect product	C₂₁₁⋊Dic₁₉

Splittings

derived, Hall	C₁₉×C₂₁₁ ⋊ C₄
Sylow	C₂₁₁ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₂₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₀₀₉
commutator, hypocenter	C₁₉×C₂₁₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₉×C₂₁₁	C₂
LCS	G ▷ C₁₉×C₂₁₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 234
transitive degree	μ_t(G) = 16036

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)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(25,234)(26,233)(27,232)(28,231)(29,230)(30,229)(31,228)(32,227)(33,226)(34,225)(35,224)(36,223)(37,222)(38,221)(39,220)(40,219)(41,218)(42,217)(43,216)(44,215)(45,214)(46,213)(47,212)(48,211)(49,210)(50,209)(51,208)(52,207)(53,206)(54,205)(55,204)(56,203)(57,202)(58,201)(59,200)(60,199)(61,198)(62,197)(63,196)(64,195)(65,194)(66,193)(67,192)(68,191)(69,190)(70,189)(71,188)(72,187)(73,186)(74,185)(75,184)(76,183)(77,182)(78,181)(79,180)(80,179)(81,178)(82,177)(83,176)(84,175)(85,174)(86,173)(87,172)(88,171)(89,170)(90,169)(91,168)(92,167)(93,166)(94,165)(95,164)(96,163)(97,162)(98,161)(99,160)(100,159)(101,158)(102,157)(103,156)(104,155)(105,154)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,147)(113,146)(114,145)(115,144)(116,143)(117,142)(118,141)(119,140)(120,139)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)

Polycyclic presentation:

2,19,211,2|a^2=d,b^a=b^18,c^a=c^210

Conjugacy classes & character table

4012 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₀₁₈
maximal normal subgroup of	Dic₈₀₁₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16036, 4);

Dic4009 #30630975