Dic₈₁₅₉ #45657644

order	32636
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₁₅₉
unambiguous semidirect product	C₁₉₉⋊Dic₄₁

Splittings

derived, Hall	C₄₁×C₁₉₉ ⋊ C₄
Sylow	C₁₉₉ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₁₉₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₁₅₉
commutator, hypocenter	C₄₁×C₁₉₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₁×C₁₉₉	C₂
LCS	G ▷ C₄₁×C₁₉₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 244
transitive degree	μ_t(G) = 32636

A minimal faithful permutation representation — 2 generators on 244 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);(1,3,2,4)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(47,244)(48,243)(49,242)(50,241)(51,240)(52,239)(53,238)(54,237)(55,236)(56,235)(57,234)(58,233)(59,232)(60,231)(61,230)(62,229)(63,228)(64,227)(65,226)(66,225)(67,224)(68,223)(69,222)(70,221)(71,220)(72,219)(73,218)(74,217)(75,216)(76,215)(77,214)(78,213)(79,212)(80,211)(81,210)(82,209)(83,208)(84,207)(85,206)(86,205)(87,204)(88,203)(89,202)(90,201)(91,200)(92,199)(93,198)(94,197)(95,196)(96,195)(97,194)(98,193)(99,192)(100,191)(101,190)(102,189)(103,188)(104,187)(105,186)(106,185)(107,184)(108,183)(109,182)(110,181)(111,180)(112,179)(113,178)(114,177)(115,176)(116,175)(117,174)(118,173)(119,172)(120,171)(121,170)(122,169)(123,168)(124,167)(125,166)(126,165)(127,164)(128,163)(129,162)(130,161)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)(145,146)

Polycyclic presentation:

2,41,199,2|a^2=d,b^a=b^40,c^a=c^198

Conjugacy classes & character table

8162 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₃₁₈
maximal normal subgroup of	Dic₁₆₃₁₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32636, 4);

Dic8159 #45657644