Dic₈₀₀₃ #44544991

order	32012
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₀₀₃
unambiguous semidirect product	C₁₅₁⋊Dic₅₃

Splittings

derived, Hall	C₅₃×C₁₅₁ ⋊ C₄
Sylow	C₁₅₁ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₁₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₀₀₃
commutator, hypocenter	C₅₃×C₁₅₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅₃×C₁₅₁	C₂
LCS	G ▷ C₅₃×C₁₅₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 208
transitive degree	μ_t(G) = 32012

A minimal faithful permutation representation — 2 generators on 208 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);(1,3,2,4)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(59,208)(60,207)(61,206)(62,205)(63,204)(64,203)(65,202)(66,201)(67,200)(68,199)(69,198)(70,197)(71,196)(72,195)(73,194)(74,193)(75,192)(76,191)(77,190)(78,189)(79,188)(80,187)(81,186)(82,185)(83,184)(84,183)(85,182)(86,181)(87,180)(88,179)(89,178)(90,177)(91,176)(92,175)(93,174)(94,173)(95,172)(96,171)(97,170)(98,169)(99,168)(100,167)(101,166)(102,165)(103,164)(104,163)(105,162)(106,161)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)(121,146)(122,145)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)

Polycyclic presentation:

2,53,151,2|a^2=d,b^a=b^52,c^a=c^150

Conjugacy classes & character table

8006 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₀₀₆
maximal normal subgroup of	Dic₁₆₀₀₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32012, 4);

Dic8003 #44544991