Dic₇₄₂₁ #44445901

order	29684
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₄₂₁

Splittings

derived, Hall	C₄₁×C₁₈₁ ⋊ C₄
Sylow	C₁₈₁ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₁₈₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₄₂₁
commutator, hypocenter	C₄₁×C₁₈₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₁×C₁₈₁	C₂
LCS	G ▷ C₄₁×C₁₈₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 226
transitive degree	μ_t(G) = 29684

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

Polycyclic presentation:

2,41,181,2|a^2=d,b^a=b^40,c^a=c^180

Conjugacy classes & character table

7424 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₈₄₂
maximal normal subgroup of	Dic₁₄₈₄₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29684, 4);

Dic7421 #44445901