Dic₈₁₃₁ #45654199

order	32524
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₁₃₁
unambiguous semidirect product	C₄₇⋊Dic₁₇₃

Splittings

derived, Hall	C₄₇×C₁₇₃ ⋊ C₄
Sylow	C₁₇₃ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₁₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₁₃₁
commutator, hypocenter	C₄₇×C₁₇₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₇×C₁₇₃	C₂
LCS	G ▷ C₄₇×C₁₇₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 224
transitive degree	μ_t(G) = 32524

A minimal faithful permutation representation — 2 generators on 224 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);(1,3,2,4)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(53,224)(54,223)(55,222)(56,221)(57,220)(58,219)(59,218)(60,217)(61,216)(62,215)(63,214)(64,213)(65,212)(66,211)(67,210)(68,209)(69,208)(70,207)(71,206)(72,205)(73,204)(74,203)(75,202)(76,201)(77,200)(78,199)(79,198)(80,197)(81,196)(82,195)(83,194)(84,193)(85,192)(86,191)(87,190)(88,189)(89,188)(90,187)(91,186)(92,185)(93,184)(94,183)(95,182)(96,181)(97,180)(98,179)(99,178)(100,177)(101,176)(102,175)(103,174)(104,173)(105,172)(106,171)(107,170)(108,169)(109,168)(110,167)(111,166)(112,165)(113,164)(114,163)(115,162)(116,161)(117,160)(118,159)(119,158)(120,157)(121,156)(122,155)(123,154)(124,153)(125,152)(126,151)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)

Polycyclic presentation:

2,47,173,2|a^2=d,b^a=b^46,c^a=c^172

Conjugacy classes & character table

8134 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₂₆₂
maximal normal subgroup of	Dic₁₆₂₆₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32524, 4);

Dic8131 #45654199