Dic₈₀₇₇ #44568165

order	32308
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₀₇₇

Splittings

derived, Hall	C₄₁×C₁₉₇ ⋊ C₄
Sylow	C₁₉₇ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₁₉₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₀₇₇
commutator, hypocenter	C₄₁×C₁₉₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₁×C₁₉₇	C₂
LCS	G ▷ C₄₁×C₁₉₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 242
transitive degree	μ_t(G) = 32308

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

Polycyclic presentation:

2,41,197,2|a^2=d,b^a=b^40,c^a=c^196

Conjugacy classes & character table

8080 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₁₅₄
maximal normal subgroup of	Dic₁₆₁₅₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32308, 4);

Dic8077 #44568165